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.In mathematics, the trigonometric functions (also called circular functions) are functions of an angle.^ Hence, they are called circular functions.
  • Sine, Cosine, and Tangent Functions 9 February 2010 13:38 UTC jwilson.coe.uga.edu [Source type: FILTERED WITH BAYES]

^ What are some examples of trigonometric circular functions?
  • WikiAnswers - What are some examples of trigonometric functions in everyday life 16 January 2010 10:48 UTC wiki.answers.com [Source type: Reference]

^ The sine function is called an odd function and so for ANY angle we have .
  • http://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx 16 January 2010 10:48 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.They are used to relate the angles of a triangle to the lengths of the sides of a triangle.^ They are used to express relations between angles and sides of triangles .
  • Sine - encyclopedia article - Citizendium 3 February 2010 18:24 UTC reid.citizendium.org [Source type: Academic]

^ If the sides of a triangle have length a, b and c with c 2 = a 2 + b 2 then the triangle has a right angle opposite the side of length c.
  • B8 Cosine Law - Vector and Complex Numbers: 9 February 2010 13:38 UTC whyslopes.com [Source type: Academic]

^ What angle, a , has a triangle that has side lengths of 1 and -1, and the length of the hypotenuse is the .
  • http://faculty.eicc.edu/bwood/ma155supplemental/supplemental7.htm 16 January 2010 10:48 UTC faculty.eicc.edu [Source type: Reference]

.Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.^ In calculus and most scientific applications, the trigonometric functions are used to model periodic phenomena (quantities that repeat).
  • Trigonometric Functions 16 January 2010 10:48 UTC euphrates.wpunj.edu [Source type: FILTERED WITH BAYES]

^ Periodicity of trigonometric functions .
  • Trigonometry - body, used, Earth, characteristics, form, waves, basic, part, Trigonometric functions, Periodicity of trigonometric functions, Applications 16 January 2010 10:48 UTC www.scienceclarified.com [Source type: Reference]

^ Trigonometric functions have many applications in algebra.

.The most familiar trigonometric functions are the sine, cosine, and tangent.^ The hyperbolic sine, cosine, and tangent .
  • Math::Trig - trigonometric functions 16 January 2010 10:48 UTC perl.enstimac.fr [Source type: FILTERED WITH BAYES]
  • Math::Trig 16 January 2010 10:48 UTC cpan.uwinnipeg.ca [Source type: FILTERED WITH BAYES]
  • Math::Trig - search.cpan.org 16 January 2010 10:48 UTC search.cpan.org [Source type: FILTERED WITH BAYES]
  • Perl 5.8 Documentation - Math::Trig - trigonometric functions 16 January 2010 10:48 UTC perl.active-venture.com [Source type: FILTERED WITH BAYES]
  • Math::Trig 12 January 2010 9:56 UTC cpan.uwinnipeg.ca [Source type: FILTERED WITH BAYES]
  • Math::Trig - trigonometric functions 12 January 2010 9:56 UTC www.xav.com [Source type: Reference]

^ Math What is sine cosine and tangent Cosine 1 : a trigonometric function that for an acute angle is the ratio betw...
  • Cosine | ChaCha Answers 9 February 2010 13:38 UTC www.chacha.com [Source type: FILTERED WITH BAYES]

^ These functions are the reciprocals of sine, cosine, and tangent, respectively: .
  • Trigonometry 16 January 2010 10:48 UTC math.brown.edu [Source type: FILTERED WITH BAYES]

.The sine function takes an angle and tells the length of the y-component (rise) of that triangle.^ Returns the trigonometric sine function of angle .

^ The reciprocal of the sine of an angle in a right triangle.
  • Cosecant - ninemsn Encarta 19 November 2009 18:15 UTC au.encarta.msn.com [Source type: FILTERED WITH BAYES]

^ The argument of the sine function is by definition an angle.
  • Waves/Sine Waves - Wikibooks, collection of open-content textbooks 3 February 2010 18:24 UTC en.wikibooks.org [Source type: Reference]

.The cosine function takes an angle and tells the length of x-component (run) of a triangle.^ If I use the cosine function, it takes an argument representing an angle in radians.
  • SQL Inverse Trig Functions - CodeCall Programming Forum 16 January 2010 10:48 UTC forum.codecall.net [Source type: General]

^ What angle, a , has a triangle that has side lengths of 1 and -1, and the length of the hypotenuse is the .
  • http://faculty.eicc.edu/bwood/ma155supplemental/supplemental7.htm 16 January 2010 10:48 UTC faculty.eicc.edu [Source type: Reference]

^ For a triangle, with sides of length a, b and c, and angle q opposite the side of length c, .
  • B8 Cosine Law - Vector and Complex Numbers: 9 February 2010 13:38 UTC whyslopes.com [Source type: Academic]

.The tangent function takes an angle and tells the slope (y-component divided by the x-component).^ At what angles is the tangent function undefined?

^ This is the same as 1 divided by the tangent of the angle.
  • Lesson BASIC TRIG FUNCTIONS 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ The Cotangent Function is equal to 1 divided by the Tangent Function.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

.More precise definitions are detailed below.^ More details on this relationship and the connection to the the slope of the tangent line to the graphs are given below.
  • The Natural Cosine and Sine Curves 9 February 2010 13:38 UTC www.math.utah.edu [Source type: FILTERED WITH BAYES]

.Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle.^ Trigonometric ratios become wave functions .

^ The basis for mensuration of triangles is the right-angled triangle.

^ Explore the Sine and Cosine trigonometric functions of a right triangle.
  • Scratch | Project | Trig Functions 16 January 2010 10:48 UTC scratch.mit.edu [Source type: FILTERED WITH BAYES]

.More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.^ The phase of a positive non-complex number is zero; that of a negative non-complex number is pi .
  • 12.5.2. Trigonometric and Related Functions 16 January 2010 10:48 UTC www.cs.cmu.edu [Source type: Reference]

^ Definition by differential equations .
  • Sine - encyclopedia article - Citizendium 3 February 2010 18:24 UTC reid.citizendium.org [Source type: Academic]

^ The absolute value of the sine of a complex number.
  • Trigonometry and Basic Functions - Numericana 16 January 2010 10:48 UTC home.att.net [Source type: Academic]

.Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles).^ Use this class of elementary functions to compute trigonometric functions and their inverses.
  • Trigonometric Functions - LabVIEW 8.5 Help 16 January 2010 10:48 UTC zone.ni.com [Source type: Reference]

^ The basis for mensuration of triangles is the right-angled triangle.

^ Explore the Sine and Cosine trigonometric functions of a right triangle.
  • Scratch | Project | Trig Functions 16 January 2010 10:48 UTC scratch.mit.edu [Source type: FILTERED WITH BAYES]

.In this use, trigonometric functions are used for instance in navigation, engineering, and physics.^ Trigonometric functions Skip to content Skip to navigation .
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Uses of trigonometric functions in the real life?
  • WikiAnswers - An example how trigonometric functions are used to explain a real life occurance 16 January 2010 10:48 UTC wiki.answers.com [Source type: General]

^ Use the graph of the trigonometric functions.
  • S-Cool | Graphs of trigonometric functions 16 January 2010 10:48 UTC www.s-cool.co.uk [Source type: Reference]

.A common use in elementary physics is resolving a vector into Cartesian coordinates.^ When we take the transform of an n -point vector using y = T x , x is decomposed into a linear combination of the basis function (rows) of T , whose coefficients are the samples of y , because x = T T y .
  • The Discrete Cosine Transform (DCT) 9 February 2010 13:38 UTC cnx.org [Source type: General]

^ Also, the sum of vectors we learn in trig class is very useful for calculus and physics.
  • Advice on trig needed for calculus and physics 16 January 2010 10:48 UTC faculty.tcc.fl.edu [Source type: Original source]

^ Assume a Cartesian system of axes with the origin in A and let the x -axis be along A–B and the y -axis be along A–D. Clearly cosα = c is the projection of e AC on the x -axis, i.e., cosα = c is the x -coordinate of this unit vector.
  • Sine - encyclopedia article - Citizendium 3 February 2010 18:24 UTC reid.citizendium.org [Source type: Academic]

.The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.^ These periodic functions can be written in sine or cosine form.

^ The arc-length parametrization of the unit circle by the cosine and sine functions .
  • The Natural Cosine and Sine Curves 9 February 2010 13:38 UTC www.math.utah.edu [Source type: FILTERED WITH BAYES]

^ Student worksheet 3 - sounds and sine waves.
  • Lesson Plan - Adding Trigonometric Functions 16 January 2010 10:48 UTC mste.illinois.edu [Source type: FILTERED WITH BAYES]

.In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another.^ What is the six type of trigonometric functions?
  • WikiAnswers - What are some examples of trigonometric functions in everyday life 16 January 2010 10:48 UTC wiki.answers.com [Source type: Reference]

^ Here are the definitions of the trigonometric functions.
  • Trigonometric Functions 16 January 2010 10:48 UTC karen.mcnabbs.org [Source type: Original source]

^ Here's how the trigonometric functions are defined.
  • Unit Circle Trigonometry 16 January 2010 10:48 UTC www.snow.edu [Source type: General]

.Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations.^ So, for instance, arcsin of X is equal to Y, this is the definition of the arcsin function.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Trigonometric functions are many-one relations.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

Contents

Right-angled triangle definitions

.
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary.
^ The basis for mensuration of triangles is the right-angled triangle.

^ These are complementary angles in a right triangle.

^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.
.The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides.^ Improve ] The obvious answer is the relationships between the sides and angles of triangles.
  • WikiAnswers - An example how trigonometric functions are used to explain a real life occurance 16 January 2010 10:48 UTC wiki.answers.com [Source type: General]

^ More information comes from the same pair of similar triangles: .
  • The Six Functions (Trig without Tears Part 2) 16 January 2010 10:48 UTC oakroadsystems.com [Source type: FILTERED WITH BAYES]

^ What angle, a , has a triangle that has side lengths of 1 and -1, and the length of the hypotenuse is the .
  • http://faculty.eicc.edu/bwood/ma155supplemental/supplemental7.htm 16 January 2010 10:48 UTC faculty.eicc.edu [Source type: Reference]

.That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same.^ What angle, a , has a triangle that has side lengths of 1 and -1, and the length of the hypotenuse is the .
  • http://faculty.eicc.edu/bwood/ma155supplemental/supplemental7.htm 16 January 2010 10:48 UTC faculty.eicc.edu [Source type: Reference]

^ And the ratios are ratios of lengths of sides of the triangle.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ As it turns out, the remaining sides of the triangle are .
  • Math Help - Trigonometry - Trigonometric Functions of Common Angles - TechnicalTutoring 16 January 2010 10:48 UTC www.hyper-ad.com [Source type: FILTERED WITH BAYES]

.If the hypotenuse is twice as long, so are the sides.^ The length of the base ( a ) is twice the side opposite to an angle a in a right triangle of hypotenuse r .
  • Trigonometry and Basic Functions - Numericana 16 January 2010 10:48 UTC home.att.net [Source type: Academic]

^ Now you will know that the sine of any angle is simply the length of the far side of the triangle (the "opposite") divided by the long side (the "hypotenuse"): .
  • Sine Function - Graph Exercise 3 February 2010 18:24 UTC www.mathsisfun.com [Source type: FILTERED WITH BAYES]

^ For example, if one side is twice as long as the other side of a rectangle, then the ratio is 2/1.
  • Geometry: geometry, sides of a right triangle, trig functions 12 January 2010 9:56 UTC en.allexperts.com [Source type: FILTERED WITH BAYES]

.It is these ratios that the trigonometric functions express.^ What are these trigonometric [trig] functions, anyway?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ These ratios define the trigonometric functions.

^ What are these trigonometric [trig] functions anyway?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.In order to define the trigonometric functions for the angle A, start with any right triangle that contains the angle A.^ To define the trigonometric functions of an angle theta assign one of the angles in a right triangle that value.
  • Trigonometry Definition Math Sheet 16 January 2010 10:48 UTC www.ecalc.com [Source type: Reference]

^ The basis for mensuration of triangles is the right-angled triangle.

^ Explore the Sine and Cosine trigonometric functions of a right triangle.
  • Scratch | Project | Trig Functions 16 January 2010 10:48 UTC scratch.mit.edu [Source type: FILTERED WITH BAYES]

The three sides of the triangle are named as follows:
.
  • The hypotenuse is the side opposite the right angle, in this case side h.^ We define the side of the triangle opposite from the right angle to be the hypotenuse, "h" .
    • Sine-Cosine-Tangent 16 January 2010 10:48 UTC wright.nasa.gov [Source type: FILTERED WITH BAYES]

    ^ Hypotenuse: The longest side of a right triangle that is opposite the right angle.
    • Trigonometry - body, used, Earth, characteristics, form, waves, basic, part, Trigonometric functions, Periodicity of trigonometric functions, Applications 16 January 2010 10:48 UTC www.scienceclarified.com [Source type: Reference]

    ^ If the sides of a triangle have length a, b and c with c 2 = a 2 + b 2 then the triangle has a right angle opposite the side of length c.
    • B8 Cosine Law - Vector and Complex Numbers: 9 February 2010 13:38 UTC whyslopes.com [Source type: Academic]

    The hypotenuse is always the longest side of a right-angled triangle.
  • The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
  • The adjacent side is the side that is in contact with (adjacent to) both the angle we are interested in (angle A) and the right angle, in this case side b.
.In ordinary Euclidean geometry, the inside angles of every triangle total 180°radians).^ As tan repeats every 180 0 or π radians we: .
  • S-Cool | Graphs of trigonometric functions 16 January 2010 10:48 UTC www.s-cool.co.uk [Source type: Reference]

^ If one adds 180 o , or π radians to A, the result will be an angle whose terminal side has the same slope as the terminal side of A. .

^ Now, if the radius of the wheel (the hypotenuse of the triangle) is 1, then the vertical line is the sine of the inside angle indicated with a red arrow.
  • Trigonometric Functions 16 January 2010 10:48 UTC www.tonmeister.ca [Source type: Original source]

.Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of 0° – 90°.^ The basis for mensuration of triangles is the right-angled triangle.

^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ So the acute angles in a right triangle are complementary.
  • Trigonometric Functions of Acute Angles 16 January 2010 10:48 UTC www.lhs.logan.k12.ut.us [Source type: Reference]

.The following definitions apply to angles in this 0° – 90° range.^ It does not work for a full range of angles from 0° to 360°, only angles between -90° and +90° will be returned, other angles will be 180° out of phase.
  • Maths - Inverse Trigonometric Functions - Martin Baker 16 January 2010 10:48 UTC www.euclideanspace.com [Source type: FILTERED WITH BAYES]

^ First, Sines and Cosines apply only to right triangles , when there is one right angle of 90 degrees, whose opposite side is called the hypotenuse.

^ Definition of Trigonometry: Trigonometry considers the properties of angles and certain ratios associated with angles, and applies the knowledge of these properties to the solution of triangles and various other algebraic and geometric problems.
  • Trigonometry Basics - Mathematics 16 January 2010 10:48 UTC mathematics.learnhub.com [Source type: FILTERED WITH BAYES]

.(They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.^ Uses and importance of functions in real life?
  • WikiAnswers - An example how trigonometric functions are used to explain a real life occurance 16 January 2010 10:48 UTC wiki.answers.com [Source type: General]

^ How can periodic functions be used in real life?
  • WikiAnswers - An example how trigonometric functions are used to explain a real life occurance 16 January 2010 10:48 UTC wiki.answers.com [Source type: General]

^ Of course, I required they not use a calculator.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

)
.The trigonometric functions are summarized in the following table and described in more detail below.^ Table for the 6 trigonometric functions for special angles.
  • Free Trigonometry Tutorials and Problems 16 January 2010 10:48 UTC www.analyzemath.com [Source type: Reference]

^ Table of Contents: trigonometric function Main .
  • trigonometric function -- Britannica Online Encyclopedia 16 January 2010 10:48 UTC www.britannica.com [Source type: Reference]

^ We summarize with the following table .
  • Integrals of Trigonometric Functions 16 January 2010 10:48 UTC www.ltcconline.net [Source type: Academic]

.The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.^ Secant of an angle = 1/Cosine of the angle = hypotenuse / adjacent.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ Cosine of an angle = adjacent / hypotenuse.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ Denote AC, the side adjacent to the angle A, by b (for base), BC, the side opposite the angle A, by a (for altitude), and the hypotenuse AB by h.
  • Trigonometry Basics - Mathematics 16 January 2010 10:48 UTC mathematics.learnhub.com [Source type: FILTERED WITH BAYES]

Function Abbreviation Description Identities (using radians)
Sine sin \frac {	extrm{opposite}} {	extrm{hypotenuse}} \sin 	heta \equiv \cos \left(\frac{\pi}{2} - 	heta \right) \equiv \frac{1}{\csc 	heta}
Cosine cos \frac {	extrm{adjacent}} {	extrm{hypotenuse}} \cos 	heta \equiv \sin \left(\frac{\pi}{2} - 	heta \right) \equiv \frac{1}{\sec 	heta}\,
Tangent tan (or tg) \frac {	extrm{opposite}} {	extrm{adjacent}} 	an 	heta \equiv \frac{\sin 	heta}{\cos 	heta} \equiv \cot \left(\frac{\pi}{2} - 	heta \right) \equiv \frac{1}{\cot 	heta}
Cotangent cot (or ctg or ctn) \frac {	extrm{adjacent}} {	extrm{opposite}} \cot 	heta \equiv \frac{\cos 	heta}{\sin 	heta} \equiv 	an \left(\frac{\pi}{2} - 	heta \right) \equiv \frac{1}{	an 	heta}
Secant sec \frac {	extrm{hypotenuse}} {	extrm{adjacent}} \sec 	heta \equiv \csc \left(\frac{\pi}{2} - 	heta \right) \equiv\frac{1}{\cos 	heta}
Cosecant csc (or cosec) \frac {	extrm{hypotenuse}} {	extrm{opposite}} \csc 	heta \equiv \sec \left(\frac{\pi}{2} - 	heta \right) \equiv\frac{1}{\sin 	heta}
.
The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle.
^ Returns the trigonometric sine function of angle .

^ The sine function is called an odd function and so for ANY angle we have .
  • http://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx 16 January 2010 10:48 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ C. do all functions in terms of sine,cosine,tangent,cosecant,secant,cotangent .
  • Outline for Teaching Trigonometry 16 January 2010 10:48 UTC www.blc.edu [Source type: Academic]

.The number θ is the length of the curve; thus angles are being measured in radians.^ III. Angle measure in radians .
  • Outline for Teaching Trigonometry 16 January 2010 10:48 UTC www.blc.edu [Source type: Academic]

^ Find the angle's radian measure.
  • http://faculty.eicc.edu/bwood/math150supnotes/preliminary7.htm 16 January 2010 10:48 UTC faculty.eicc.edu [Source type: Academic]

^ The Sine Curve y = Sin( x ) where y is measured in radians .
  • Unit Circle: Sine and Cosine Functions 9 February 2010 13:38 UTC curvebank.calstatela.edu [Source type: FILTERED WITH BAYES]

.The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line.^ The sine function is one of the basic functions encountered in trigonometry (the others being the cosecant , cosine , cotangent , secant , and tangent ).

^ Most students will be familiar with the right triangle definitions of these functions (sine, cosine and tangent are abbreviated sin, cos, and tan, respectively): Right triangle geometry.
  • Trigonometry and Angles 12 January 2010 9:56 UTC serc.carleton.edu [Source type: FILTERED WITH BAYES]

^ Sine.svg Cosine.svg Tangent.svg Cotangent.svg Secant.svg Cosecant.svg .
  • File:Cosine.svg - Wikimedia Commons 9 February 2010 13:38 UTC commons.wikimedia.org [Source type: Reference]

("Fixed" in this context means not moving as θ changes; "moving" means depending on θ.) Thus, as θ goes from 0 up to a right angle, sin θ goes from 0 to 1, tan θ goes from 0 to ∞, and sec θ goes from 1 to ∞.
.
The cosine, cotangent, and cosecant functions of an angle θ constructed geometrically in terms of a unit circle.
^ Returns the trigonometric cosine function of angle .

^ Arbitrary angles and the unit circle .
  • Trigonometric Functions 16 January 2010 10:48 UTC www.clarku.edu [Source type: FILTERED WITH BAYES]

^ Special angles together with their sine and cosine displayed on a unit circle.
  • Free Trigonometry Tutorials and Problems 16 January 2010 10:48 UTC www.analyzemath.com [Source type: Reference]

The functions whose names have the prefix co- use horizontal lines where the others use vertical lines.

Sine

.The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.^ Students learn that the sine of an angle of a right triangle is equal to the length of the side opposite the angle over t...
  • YouTube - Watch Video on Sine Cosine Tangent - SOHCAHTOA 9 February 2010 13:38 UTC www.youtube.com [Source type: FILTERED WITH BAYES]

^ Similarly, the cosine of angle A equals the ratio of the adjacent side over the hypotenuse.

^ In a right triangle, the ratio of the length of the leg opposite an angle to the length of the hypotenuse is called the sine of the angle.
  • Exploration Guide: Sine, Cosine and Tangent Gizmo | ExploreLearning 9 February 2010 13:38 UTC www.explorelearning.com [Source type: General]

In our case
\sin A = \frac {	extrm{opposite}} {	extrm{hypotenuse}} = \frac {a} {h}.
.Note that this ratio does not depend on size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ So the acute angles in a right triangle are complementary.
  • Trigonometric Functions of Acute Angles 16 January 2010 10:48 UTC www.lhs.logan.k12.ut.us [Source type: Reference]

^ How to define the cosine ratio and identify the cosine of an angle in a right triangle.

Cosine

.The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.^ Suppose we know the angle C and the sides adjacent to it, a and b.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ The side adjacent to q has length .
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

^ The adjacent side cannot be the hypotenuse.
  • Trig: Sine, Cosine, and Tangent 215 - Shop Essentials Training 9 February 2010 13:38 UTC www.toolingu.com [Source type: Reference]

In our case
\cos A = \frac {	extrm{adjacent}} {	extrm{hypotenuse}} = \frac {b} {h}.

Tangent

.The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.^ This ratio is positive or negative depending on the signs of the adjacent and opposite sides.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ The signs of the adjacent and opposite sides are dependent on the quadrant that the angle is in.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ Similarly, the cosine of angle A equals the ratio of the adjacent side over the hypotenuse.

In our case
	an A = \frac {	extrm{opposite}} {	extrm{adjacent}} = \frac {a} {b}.

Reciprocal functions

.The remaining three functions are best defined using the above three functions.^ Khan Academy Presents: Using the unit circle to define the sine, cosine, and tangent functions (10:11) .
  • Shahrukh Khan And His Forthcoming Film “My Name Is Khan” - Astrological Analysis 9 February 2010 13:38 UTC www.articlesbase.com [Source type: General]

^ The solution is left for the next chapter that will go into detail to define the Sine function mathematically by using a circle and arclength.

^ Use all three built-in trig functions, sin, cos, and atan, with a variety of positive and negative numbers until you are comfortable "thinking" in radians.
  • Lesson 7, Trig functions 16 January 2010 10:48 UTC ronleigh.info [Source type: FILTERED WITH BAYES]

The cosecant csc(A), or cosec(A), is the reciprocal of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
\csc A = \frac {	extrm{hypotenuse}} {	extrm{opposite}} = \frac {h} {a}.
The secant sec(A) is the reciprocal of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
\sec A = \frac {	extrm{hypotenuse}} {	extrm{adjacent}} = \frac {h} {b}.
The cotangent cot(A) is the reciprocal of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:
\cot A = \frac {	extrm{adjacent}} {	extrm{opposite}} = \frac {b} {a}.

Slope definitions

.Equivalent to the right-triangle definitions the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line.^ The trigonometric functions are most simply defined using the right triangle.

^ The complex trigonometric functions can be defined algebraically in terms of complex exponentials as: .

^ Trigonometric ratios are defined for acute angle in a right angle triangle.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

The slope is commonly taught as "rise over run" or rise/run. .The three main trigonometric functions are commonly taught in the order sine, cosine, tangent.^ They provide Tangent Function, Sine Function, and Cosine Function at least.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ So these are the inverse trig functions for sine, cosine and tangent.
  • Calculus: Inverse Sine, Cosine, and Tangent | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: General]

^ The arctangent function is inverse function of trigonometric tangent function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

With a unit circle, the following correspondence of definitions exists:
  1. Sine is first, rise is first. .Sine takes an angle and tells the rise when the length of the line is 1.
  2. Cosine is second, run is second.^ This worksheet gives you practice at using the sine ratio to calculate the length of unknown sides and the size of angles in a right angled triangle.The sine ratio is the opposite side from the a...
    • Documents Tagged with sine | Scribd 3 February 2010 18:24 UTC www.scribd.com [Source type: General]

    ^ Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle.
    • Trigonometric Functions Help : Videos | Worksheets | Word Problems 16 January 2010 10:48 UTC tulyn.com [Source type: FILTERED WITH BAYES]

    ^ For remembrance sake it can also be written as; This useful identity tells us that even the Cosine of an angle can be expressed in terms of the Sine of the angle, therefore the Sine function is the fundamental Trigonometric function, as all other functions can be derived from it, including the Cosine function.

    .Cosine takes an angle and tells the run when the length of the line is 1.
  3. Tangent is the slope formula that combines the rise and run.^ What I'd like for us to do is take a look at what the slopes of the tangent lines are here.
    • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]
    • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

    ^ Students create a large unit circle for later reference, labeled with common angles, their coordinates (cosine and sine), and their slopes (tangent) NYates published this 05 / 25 / 2009 129 reads 0 comments 5 p.
    • Documents Tagged with sine | Scribd 3 February 2010 18:24 UTC www.scribd.com [Source type: General]

    ^ The Sine Rule states that the ratio of any side in a triangle divided by the sine of the opposite angle equals a constant.The Cosine Rule is an equation for the length of a side expressed in terms ...
    • Documents Tagged with sine | Scribd 3 February 2010 18:24 UTC www.scribd.com [Source type: General]

    Tangent takes an angle and tells the slope, and tells the rise when the run is 1.
.This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e., angles and slopes.^ There's actually an easy way to convert between the two.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ If you plan to use gradients, also convert the angles to gradients.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ Line 2 uses the inverse sine method of the Math object (which returns an angle in radians) and then converts it to degrees.
  • Flash Game Design: Trigonometry 101 > The Heart of Trig 16 January 2010 10:48 UTC www.adobepress.com [Source type: FILTERED WITH BAYES]
  • Peachpit: Flash Game Design: Trigonometry 101 > The Heart of Trig 16 January 2010 10:48 UTC www.peachpit.com [Source type: General]

.(Note that the arctangent or "inverse tangent" is not to be confused with the cotangent, which is cos divided by sin.^ The arctangent function is inverse function of trigonometric tangent function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The Cotangent Function is equal to 1 divided by the Tangent Function.
  • Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES 16 January 2010 10:48 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ Just a note (it says this in the post too) that wrapping FastMath in a function call negates the speed increase, and you get the same results as just using Math.sin/cos.
  • polygonal labs » Fast and accurate sine/cosine approximation 9 February 2010 13:38 UTC lab.polygonal.de [Source type: General]

)
.While the radius of the circle makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run.^ Does the type of button make any difference?
  • Apple - Support - Discussions - Problem with trigonometric functions ... 16 January 2010 10:48 UTC discussions.apple.com [Source type: General]

^ This unit also is running with 3GB of RAM and I’m sure that makes a difference as well.
  • GBM Review: The HP 2730p Elitebook Tablet PC | GottaBeMobile.com 12 January 2010 9:56 UTC www.gottabemobile.com [Source type: General]

^ No matter how clunky you make an expression, it makes no difference if, at the end of the day, it amounts to a simple parabola.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

.To adjust and find the actual rise and run, just multiply the sine and cosine by the radius.^ For instance, find the sine and cosine of 10 o , then of -10 o .
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

^ But, look it here, I can actually take this, just to prove it to you, these are actually the same thing, I can take this fraction and multiply it by square root of two over square root of two, I am just multiplying by one, square root of two over square root of two is this one.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ As the radius rotates, the trigonometric functions go through a regular cycle of variations with period 2π, the sine and cosine bounded by ±1, the tangent and cotangent going to ±∞.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

For instance, if the circle has radius 5, the run at an angle of 1° is 5 cos(1°)

Unit-circle definitions

.The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin.^ Trigonometric functions are many-one relations.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The hypotenuse is the radius of the circle, which is one.
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

^ A circle centered in O and with radius = 1, is called a trigonometric circle or unit circle.
  • An introduction to TRIGONOMETRY 16 January 2010 10:48 UTC www.ping.be [Source type: Academic]

.The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ To the angle of the triangle that is at the center of the circle.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Right triangle and unit-circle definitions .

.The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 radians.^ Find all the trigonometric functions of the angle A opposite the side 4.
  • Trigonometry Basics - Mathematics 16 January 2010 10:48 UTC mathematics.learnhub.com [Source type: FILTERED WITH BAYES]

^ All positive rationals (and their square roots) as trigonometric functions of zero!
  • Trigonometry and Basic Functions - Numericana 16 January 2010 10:48 UTC home.att.net [Source type: Academic]

^ Trigonometric tables of all 6 trigonometric functions, with angles in degrees and radians.
  • Free Trigonometry Tutorials and Problems 16 January 2010 10:48 UTC www.analyzemath.com [Source type: Reference]

.It also provides a single visual picture that encapsulates at once all the important triangles.^ The triangle picture of trig is very useful, but once we have it, we can discuss the trigonometric functions in a number of other useful ways.
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

^ Nova Creations' DialogSedan is a most varsatile component provides all under one roof solution for Visual Basic Software developers.
  • Trigonometric functions downloads at VicMan 16 January 2010 10:48 UTC www.vicman.net [Source type: Reference]

^ Each picture can be used to tell you different things about the trigonometric functions, but it's important to remember that all three pictures are describing the same sine, cosine, and tangent.
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

From the Pythagorean theorem the equation for the unit circle is:
x^2 + y^2 = 1. \,
.In the picture, some common angles, measured in radians, are given.^ Let b be the radian measure of angle KOJ .
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Remember that 1.2 is given in radian measure.
  • Evaluating Trigonometric Functions 12 January 2010 9:56 UTC www.www-mathtutor.com [Source type: Reference]

^ Let a be the radian measure of angle MOK .
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

.Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles.^ The measurement of angle in anticlockwise direction is considered positive and negative in clockwise direction.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ What does a negative angle (measure) mean?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ The ordinate of the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

.Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle.^ Let v be the image point corresponding with the angle pi/4 on the unit circle.
  • An introduction to TRIGONOMETRY 16 January 2010 10:48 UTC www.ping.be [Source type: Academic]

^ So to start with, let's look at the unit circle.
  • Trigonometry or Trig: Math 16 January 2010 10:48 UTC www4.ncsu.edu [Source type: Original source]

^ Take an x-axis and an y-axis (orthonormal) and let O be the origin.
  • An introduction to TRIGONOMETRY 16 January 2010 10:48 UTC www.ping.be [Source type: Academic]

.The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively.^ Again referring to Figure 0.47 , we define sin to be the y- coordinate of the point on the circle and cos to be the x - coordinate of the point.

^ First, it is equal to sin a sin b cos C from evaluating the cross products.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ I use fixed point math and lookup tables for sin, cos and atan.
  • BREW Forums - about trigonometric function 16 January 2010 10:48 UTC brewforums.qualcomm.com [Source type: FILTERED WITH BAYES]

The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.
Note that these values can easily be memorized in the form
	frac{1}{2}\sqrt{0},\quad 	frac{1}{2}\sqrt{1},\quad 	frac{1}{2}\sqrt{2},\quad 	frac{1}{2}\sqrt{3},\quad 	frac{1}{2}\sqrt{4}.

The sine and cosine functions graphed on the Cartesian plane.
For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine and cosine are periodic functions with period 2π:
\sin	heta = \sin\left(	heta + 2\pi k \right),\,
\cos	heta = \cos\left(	heta + 2\pi k \right),\,
for any angle θ and any integer k.
.The smallest positive period of a periodic function is called the primitive period of the function.^ The period is and is the smallest time t needed for the function to execute on complete oscillation or cycle.
  • Sine Waves and Sound 3 February 2010 18:24 UTC mathdemos.gcsu.edu [Source type: Reference]

^ The smallest such number T > 0 is called the fundamental period .

^ The trig function period identities are [X is an angle, n is a positive integer]: .
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.
Trigonometric functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent
Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:
 \begin{align} 	an	heta & = \frac{\sin	heta}{\cos	heta},\ \cot	heta = \frac{\cos	heta}{\sin	heta} = \frac{1}{	an	heta} \\[10pt] \sec	heta & = \frac{1}{\cos	heta},\ \csc	heta = \frac{1}{\sin	heta} \end{align}
So :
  • The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.
  • The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.
.To the right is an image that displays a noticeably different graph of the trigonometric function f(θ)= tan(θ) graphed on the Cartesian plane.^ The Trigonometric Functions Graphing Sine and Cosine Functions Graphing Sine or Cosine Functions with Different Coefficients Page [1 of 3] Let’s see how we get to look at variations on the theme.
  • Pre-Calculus: Graph Sine, Cosine with Coefficients | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: Original source]

^ For any (x, y) in the Cartesian product of Complex by Complex, from the foregoing real addition theorems, with the replacement of y by i y and the use of the circular trigonometric functions , the following complex addition theorems are obvious .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ This interval is easily visible on graphs of the corresponding trigonometric function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

  • Note that its x-intercepts correspond to that of sin(θ) while its undefined values correspond to the x-intercepts of the cos(θ).
  • Observe that the function's results change slowly around angles of kπ, but change rapidly at angles close to (k + 1/2)π.
  • The graph of the tangent function also has a vertical asymptote at θ = (k + 1/2)π.
  • This is the case because the function approaches infinity as θ approaches (k + 1/2)π from the left and minus infinity as it approaches (k + 1/2)π from the right.

All of the trigonometric functions of the angle θ can be constructed geometrically in terms of a unit circle centered at O.
.Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (as shown in the picture to the right), and similar such geometric definitions were used historically.^ As such, we can not define an inverse of trigonometric function in the first place!
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ Trigonometric functions : Memorize a simple picture for 3 basic definitions.
  • Trigonometry and Basic Functions - Numericana 16 January 2010 10:48 UTC home.att.net [Source type: Academic]

^ The ordinate of the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

.
  • In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India[1] (see history).
  • cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD.
  • tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function.^ It is easy to see that the horizontal distance is s cos θ, where θ is the angle of depression.
    • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

    ^ Hence, domain and range of tangent function are : .
    • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

    ^ The length of the base of triangle ACO (that is the line segment, OC ) is shown as k .
    • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

    cot(θ) is another tangent segment, .AF.
  • sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.
  • DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle).
  • From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero.^ This is explicitly in the function names: cosine is the cofunction of sine, cotangent is the cofunction of tangent, and cosecant is the cofunction of secant.
    • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

    ^ The tangent line is horizontal at that point, the derivative is zero.
    • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]
    • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

    ^ It is easy to see that the horizontal distance is s cos θ, where θ is the angle of depression.
    • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

    .(Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.^ Trigonometric functions covers the concepts, formulas, and graphs used in trigonometry, and introduces some of the basic identities .
    • Trigonometric Functions - Trigonometry - Brightstorm 16 January 2010 10:48 UTC www.brightstorm.com [Source type: Reference]

    [2])

Series definitions

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.
.Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine.^ So in fact, we see that a good guess for the derivative of cosine is negative sine.
  • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

^ So the derivative of cosine is not sine, but is instead, negative sine.
  • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

^ And the range of sine and cosine goes from -1 to 1 only.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

.(Here, and generally in calculus, all angles are measured in radians; see also the significance of radians below.^ Let b be the radian measure of angle KOJ .
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Let a be the radian measure of angle MOK .
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ For angles, this means that whether you're working in radians, degrees or brads, they'll all result in the same circle-fraction, z .
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

) One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:[3]
 \begin{align} \sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt] & = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}, \\[8pt] \cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\[8pt] & = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}. \end{align}
.These identities are sometimes taken as the definitions of the sine and cosine function.^ So these are the inverse trig functions for sine, cosine and tangent.
  • Calculus: Inverse Sine, Cosine, and Tangent | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: General]

^ Elementary Functions and Their Inverses Inverse Trigonometric Functions The Inverse Sine, Cosine and Tangent Functions Page [1 of 2] Okay, so let’s take a look at the inverse functions for the standard trigonometric functions.
  • Calculus: Inverse Sine, Cosine, and Tangent | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: General]

^ Professor Burger will teach you how to calculate the derivative of the most common trig functions (sine, cosine, and tangent).
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]
  • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

.They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g., in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations.^ Take the infinite Geometric series .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ Since they son't really have to line up the numbers for multiplication.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ They are inverse functions corresponding to trigonometric functions.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

.The differentiability and continuity of these functions are then established from the series definitions alone.^ This is hard to do in a trig/calc heavy curriculum because continuous differentiable real valued functions are complicated.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ Nobody ever thinks it unusual that high school is dedicated, essentially, to the study of continuous and differentiable real-valued functions?
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ Since these ratios depend upon the angle for their values, they are the functions of the angle according to the general definition of a function that we discussed at the beginning of our lesson.
  • Trigonometry Basics - Mathematics 16 January 2010 10:48 UTC mathematics.learnhub.com [Source type: FILTERED WITH BAYES]

.Combining these two series gives Euler's formula: cos x + i sin x = eix.^ If we write sin b sin c cos A = cos a - cos b cos c by rearranging the formula for cos a, and then substitute the formula for cos c in this, we can use cos 2 b = 1 - sin 2 b and then divide by sin b to find sin c cos A = cos a sin b - cos b sin a cos C. This formula can be used to find c after using the cot formula to find A in an SAS case, instead of using the Law of Sines directly.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ Equating these, we find cos c = cos a cos b + sin a sin b cosC, which is the Law of Cosines for the side c.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ This formula gives zero when the length of one side is the same as the length of the other two sides, but will malfunction when one side has length zero.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

Other series can be found.[4] For the following trigonometric functions:
Un is the nth up/down number,
Bn is the nth Bernoulli number, and
En (below) is the nth Euler number.
Tangent
 \begin{align} 	an x & {} = \sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!} \\[8pt] & {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} x^{2n-1}}{(2n)!} \\[8pt] & {} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315} + \cdots, \qquad 	ext{for } |x| < \frac{\pi}{2}. \end{align}
.When this series for the tangent function is expressed in a form in which the denominators are the corresponding factorials, and the numerators, called the "tangent numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of odd cardinality.^ Although all the answers seem correct, they are expressions, minomials, operations, and EXPRESS the value of a number, but they are not themselves numbers.
  • Addicted | Iranian.com 12 January 2010 9:56 UTC www.iranian.com [Source type: General]

^ Because any angle that we put in, no matter if we go on to infinity or negative infinity, any angle we put in is going to go into this function, we call the sine function and it is going to spit out a number, that is always going to be between this.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Another property of symmetrical functions is that, if you break them down into series expansions, odd functions will only have odd terms, and even functions only have even terms.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

[5]
Cosecant
 \begin{align} \csc x & {} = \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n} x^{2n-1}}{(2n)!} \\[8pt] & {} = \frac {1} {x} + \frac {x} {6} + \frac {7 x^3} {360} + \frac {31 x^5} {15120} + \cdots, \qquad 	ext{for } 0 < |x| < \pi. \end{align}
Secant
 \begin{align} \sec x & {} = \sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!} = \sum_{n=0}^\infty \frac{(-1)^n E_{2n} x^{2n}}{(2n)!} \\[8pt] & {} = 1 + \frac {x^2} {2} + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \cdots, \qquad 	ext{for } |x| < \frac{\pi}{2}. \end{align}
.When this series for the secant function is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "secant numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of even cardinality.^ Because any angle that we put in, no matter if we go on to infinity or negative infinity, any angle we put in is going to go into this function, we call the sine function and it is going to spit out a number, that is always going to be between this.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Another property of symmetrical functions is that, if you break them down into series expansions, odd functions will only have odd terms, and even functions only have even terms.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ This is not the only way of defining trigonometric functions; they can be defined as analytic functions of a complex variable z by power series, for example.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

[citation needed]
Cotangent
 \begin{align} \cot x & {} = \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!} \\[8pt] & {} = \frac {1} {x} - \frac {x}{3} - \frac {x^3} {45} - \frac {2 x^5} {945} - \cdots, \qquad 	ext{for } 0 < |x| < \pi. \end{align}
.From a theorem in complex analysis, there is a unique analytic continuation of this real function to the domain of complex numbers.^ For a real number “x”, there is a secant function defined as : .
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ For a real number “x”, there is a cosecant function defined as : .
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ For a real number “x”, there is a tangent function defined as : .
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

.They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.^ This is not the only way of defining trigonometric functions; they can be defined as analytic functions of a complex variable z by power series, for example.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ They are inverse functions corresponding to trigonometric functions.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The class containing the functions must also define a method named getNmbr(), which takes no parameters and returns the number of functions to be plotted.
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

Relationship to exponential function and complex numbers

Euler's formula illustrated with the three dimensional helix, starting with the 2-D orthogonal components of the unit circle, sine and cosine (using θ = t ).
It can be shown from the series definitions[6] that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:
 e^{i 	heta} = \cos	heta + i\sin	heta. \,
This identity is called Euler's formula. .In this way, trigonometric functions become essential in the geometric interpretation of complex analysis.^ For any (x, y) in the Cartesian product of Complex by Complex, from the foregoing real addition theorems, with the replacement of y by i y and the use of the circular trigonometric functions , the following complex addition theorems are obvious .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ Now, first of all, the question that we have to ask is, “Do trigonometric functions have inverses?” Well, there’s one way to find out.
  • Calculus: Inverse Sine, Cosine, and Tangent | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: General]

^ This is not the only way of defining trigonometric functions; they can be defined as analytic functions of a complex variable z by power series, for example.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

.For example, with the above identity, if one considers the unit circle in the complex plane, defined by e ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.^ Trigonometric functions are many-one relations.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The exponent of the exponential function is inverse trigonometric function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The arccosine function is inverse function of trigonometric cosine function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:
\sin z = \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \frac{e^{i z} - e^{-i z}}{2i}\, = \frac{\sinh \left( i z\right) }{i}
\cos z = \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n} = \frac{e^{i z} + e^{-i z}}{2}\, = \cosh \left(i z\right)
where i 2 = −1. Also, for purely real x,
\cos x = \mbox{Re } (e^{i x}) \,
\sin x = \mbox{Im } (e^{i x}) \,
.It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ This means that argument of sine function is "R".
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Solution : The cosine function is valid for all real values of its argument.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

\sin (x + iy) = \sin x \cosh y + i \cos x \sinh y,\,
\cos (x + iy) = \cos x \cosh y - i \sin x \sinh y.\,
.This exhibits a deep relationship between the complex sine and cosine functions and their real and real hyperbolic counterparts.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Well, what’s going to happen is The Trigonometric Functions Graphing Sine and Cosine Functions Graphing Sine or Cosine Functions with Different Coefficients Page [2 of 3] all the x values get doubled.
  • Pre-Calculus: Graph Sine, Cosine with Coefficients | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: Original source]

^ Now in fact, if you really wanted to figure out for example, the derivative of the sine function, let's start with sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]
  • Calculus: Derivatives of Trigonometric Functions | how-to videos powered by MindBites 16 January 2010 10:48 UTC thinkwell.mindbites.com [Source type: Original source]

Complex graphs

.In the following graphs, the domain is the complex plane pictured, and the range values are indicated at each point by color.^ Therefore, it follows that domain and range of trigonometric function are exchanged for corresponding inverse function i.e.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ Because a function can assign no more than one range-value to each domain-value.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Since each of these comes back to any given y -value in its range infinitely many times, you can't just transpose their graphs to form their inverses.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

Brightness indicates the size (absolute value) of the range value, with black being zero. .Hue varies with argument, or angle, measured from the positive real axis.^ Even for angles, which are not acute, we consider trigonometric ratios as ratios of sides or ratios of a side and hypotenuse of the right angle triangle OAB, which is constructed with the terminal ray, “OA” (measuring angle from the initial position in x-direction) and x-axis.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The measurement of angle in anticlockwise direction is considered positive and negative in clockwise direction.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ In order to find, domain of the function, we need to find values of “x” for which argument of logarithmic function is a positive real number.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

(more)
Trigonometric functions in the complex plane
Complex sin.jpg
Complex cos.jpg
Complex tan.jpg
Complex Cot.jpg
Complex Sec.jpg
Complex Csc.jpg
 \sin z\,  \cos z\,  	an z\,  \cot z\,  \sec z\,  \csc z\,

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation
y'' = -y.\,
That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional function space V consisting of all solutions of this equation,
  • the sine function is the unique solution satisfying the initial condition \scriptstyle \left( y'(0), y(0) \right) = (1, 0)\, and
  • the cosine function is the unique solution satisfying the initial condition \scriptstyle \left( y'(0), y(0) \right) = (0, 1)\,.
.Since the sine and cosine functions are linearly independent, together they form a basis of V.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ So, what we did was we wrote it in terms of the function, use sine and cosine because that helps you walk around the circle and plug in values.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Well, what’s going to happen is The Trigonometric Functions Graphing Sine and Cosine Functions Graphing Sine or Cosine Functions with Different Coefficients Page [2 of 3] all the x values get doubled.
  • Pre-Calculus: Graph Sine, Cosine with Coefficients | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: Original source]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

(See linear differential equation.) .It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions.^ And the range of sine and cosine goes from -1 to 1 only.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Well, what’s going to happen is The Trigonometric Functions Graphing Sine and Cosine Functions Graphing Sine or Cosine Functions with Different Coefficients Page [2 of 3] all the x values get doubled.
  • Pre-Calculus: Graph Sine, Cosine with Coefficients | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: Original source]

.Further, the observation that sine and cosine satisfies y′′ = −y means that they are eigenfunctions of the second-derivative operator.^ So the derivative of cosine is not sine, but is instead, negative sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

^ So in fact, we see that a good guess for the derivative of cosine is negative sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

^ Professor Burger will teach you how to calculate the derivative of the most common trig functions (sine, cosine, and tangent).
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

The tangent function is the unique solution of the nonlinear differential equation
y' = 1 + y^2\,
satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see Needham's Visual Complex Analysis.[7]

The significance of radians

.Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ This means that argument of sine function is "R".
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The angle measure in radians is simply arc length on a circle of radius 1.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them.^ And the range of sine and cosine goes from -1 to 1 only.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Right now, go set your calculator so that it computes sines and cosines of angles given in radians, and then leave it that way.
  • Karl's Calculus Tutor - May the Circle be Unbroken (trig functions) 16 January 2010 10:48 UTC www.karlscalculus.org [Source type: FILTERED WITH BAYES]

^ Substitute back into the equations for the sine and cosine of theta, to obtain .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

If an argument to sine or cosine in radians is scaled by frequency,
f(x) = \sin kx, \,
then the derivatives will scale by amplitude.
f'(x) = k\cos kx. \,
Here, k is a constant that represents a mapping between units. If x is in degrees, then
k = \frac{\pi}{180^\circ}.
This means that the second derivative of a sine in degrees satisfies not the differential equation
y'' = -y\,
but rather
y'' = -k^2 y.\,
The cosine's second derivative behaves similarly.
.This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.^ The built-in functions only take radian arguments.
  • Trigonometric Functions - Rosetta Code 16 January 2010 10:48 UTC rosettacode.org [Source type: Reference]

^ So these are the inverse trig functions for sine, cosine and tangent.
  • Calculus: Inverse Sine, Cosine, and Tangent | MindBites.com 9 February 2010 13:38 UTC www.mindbites.com [Source type: General]

^ So the derivative of cosine is not sine, but is instead, negative sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

Identities

.Many identities exist which interrelate the trigonometric functions.^ Trigonometric functions are many-one relations.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ We can identify many such shortened intervals for a particular trigonometric function.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and definitions .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem.^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ This is the generalization of the Pythagorean theorem to non-right triangles.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ That's sine squared x, all over cosine squared x.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

In symbolic form, the Pythagorean identity is written
\sin^2 x + \cos^2 x = 1, \,
where sin2 x + cos2 x is standard notation for (sin x)2 + (cos x)2.
.Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves.^ What are the angle sum, difference, and/or doubling formulae?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ These formulae may be derived from the angle sum and difference formulae.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ It's the sum formula for the sine function.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

.These can be derived geometrically, using arguments which go back to Ptolemy; one can also produce them algebraically using Euler's formula.^ You can add one by going in-game and following these easy instructions.
  • EverQuest II Sentinel's Fate - Massively Multiplayer Role-Playing Game 9 February 2010 13:38 UTC eq2players.station.sony.com [Source type: FILTERED WITH BAYES]

^ One thing I just want to point out to you here is you are going to have to memorize these things.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y, \,
\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y, \,
\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y, \,
\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y. \,
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
.These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations, much like the logarithm function.^ The sum of two numbers is 15.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ What is the sum of the reciprocals of the two numbers?
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ The product of the same two numbers is 16.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

Calculus

.For integrals and derivatives of trigonometric functions, see the relevant sections of List of differentiation identities, Lists of integrals and List of integrals of trigonometric functions.^ The Riemann Integral is an ant-derivative, thus each of these integral formulae may be verified by differentiation.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ In this section of Advanced Calculus 2, we continue to cover calculus topics, and learn about the use of inverse trigonometric functions.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ The Derivatives of Trigonometric Functions The derivatives of trigonometric functions - Professor Edward Burger explains derivatives of trigonometric functions in this video from Thinkwell's online Calculus series.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.Below is the list of the derivatives and integrals of the six basic trigonometric functions.^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ Some relations between trigonometric functions are an immediate consequence of the definitions, such as cos 2 x + sin 2 x = 1, but many are not quite so easy to derive, though very useful.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ Thus, we have obtained the first of the derivative formulae of the inverse hyperbolic trigonometric functions .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

The number C is a constant of integration.
\ \ \ \ f(x) \ \ \ \ f'(x) \int f(x)\,dx
\,\ \sin x \,\ \cos x \,\ -\cos x + C
\,\ \cos x \,\ -\sin x \,\ \sin x + C
\,\ 	an x \,\ \sec^2 x = 1+	an^2 x -\ln \left |\cos x\right | + C
\,\ \cot x \,\ -\csc^2 x = -(1+\cot^2 x) \ln \left |\sin x\right | + C
\,\ \sec x \,\ \sec x	an x \ln \left |\sec x + 	an x\right | + C
\,\ \csc x \,\ -\csc x \cot x \ -\ln \left |\csc x + \cot x\right | + C

Definitions using functional equations

.In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the sum and difference formulas.^ What are the angle sum, difference, and/or doubling formulae?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ It's the sum formula for the sine function.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

^ Substitute in the angle sum and difference formulae for the affected functions, and simplify.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.Taking as given these formulas and the Pythagorean identity, for example, one can prove that only two real functions satisfy those conditions.^ The given function is real for all real values of x.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ In the case of real t, this formula yields only one branch of the hyperbola.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ Now in fact, if you really wanted to figure out for example, the derivative of the sine function, let's start with sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

Symbolically, we say that there exists exactly one pair of real functions — \scriptstyle \sin\, and \scriptstyle \cos\, — such that for all real numbers \scriptstyle x\, and \scriptstyle y\,, the following equations hold:[citation needed]
\sin^2 x + \cos^2 x = 1\,
\sin(x\pm y) = \sin x\cos y \pm \cos x\sin y\,
\cos(x\pm y) = \cos x\cos y \mp \sin x\sin y\,
with the added condition that \scriptstyle 0 < x\cos x < \sin x < x\, for \scriptstyle 0 < x < 1\,.
.Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers.^ Now in fact, if you really wanted to figure out for example, the derivative of the sine function, let's start with sine.
  • Calculus: Derivatives of Trigonometric Functions | MindBites.com 16 January 2010 10:48 UTC www.mindbites.com [Source type: General]

^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ It is not easy to find other functions with such a limited table.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

.As an example, this derivation can be used to define trigonometry in Galois fields.^ However, it is useful in those cases where it is necessary to refresh the values of instance variables defined in the class (such as a counter, for example).
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

^ For example, instead using the derivative at ½π, I could have used it at x = 0.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

Computation

.The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle.^ Because any angle that we put in, no matter if we go on to infinity or negative infinity, any angle we put in is going to go into this function, we call the sine function and it is going to spit out a number, that is always going to be between this.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ So, this angle that you picked with that thinking to begin with was negative pi over six and it worked, but it is just not legal for what the inverse cotangent function can return because you have to get an angle between zero and pi.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Notice that for the cotangent function, which was the original function, if you look back four for cotangent, the angles that you need to return was zero to pi and notice that this did fall between zero and pi because here is zero and here is pi, so this angle did fall where we were.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found.^ For serious computation, tables such as Vega's (see References) were used.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ This is the entire and exclusive Agreement between you and us regarding use of the Site and it cannot be modified, except as specifically described below in Section 2.
  • Trigonometric Identities | SPIKE 16 January 2010 10:48 UTC www.spike.com [Source type: General]

^ Here, we shall apply these ratios in the context of any real value angle, represented on a real number line.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

.The first step in computing any trigonometric function is range reduction — reducing the given angle to a "reduced angle" inside a small range of angles, say 0 to π/2, using the periodicity and symmetries of the trigonometric functions.^ We say that all trig functions have a period.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ The nature of trigonometric functions is periodic.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ Hence, range of given function is a singleton : .
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

.Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures.^ Trigonometric functions are many-one relations.
  • Inverse trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: Academic]

^ The seven-place tables give tabulated values for 5 significant figures.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ However, it may be more accurate to subtract the correction s vers θ, since this is usually a small number and need not be calculated to the full number of significant figures required.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

.Such tables have been available for as long as trigonometric functions have been described (see History below), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2) = 1).^ SFG See Shepard function generator .
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

^ Review the available sources of functional values.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ Angle classifications Angle relations Arc length from central angle and radius Cofunction relations Supplementary angle relations Algebraically even and odd trigonometric functions Basic reference tables Trigonometric function evaluation from sides of a right triangle Trigonometric function signs from quadrants of the unit circle Trigonometric functions that are multiplicative inverses of each other Trigonometric functions in terms of sin(X) and cos(X) Pythagorean Theorem Pythagorean Identities .
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

Modern computers use a variety of techniques.[8] .One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction.^ For serious computation, tables such as Vega's (see References) were used.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ They end up with the money not you.
  • Celebrex and Bextra Lawsuit Settlements Reported - AboutLawsuits.com 12 January 2010 9:56 UTC www.aboutlawsuits.com [Source type: General]

^ Only when radians are used is the sine of an angle approximately equal to the angle for small angles.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

[9] .On devices that lack hardware multipliers, an algorithm called CORDIC (as well as related techniques) which uses only addition, subtraction, bitshift and table lookup, is often used.^ "The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers .
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

^ We will only approve comments that are directly related to the blog, use appropriate language and are not attacking the comments of others.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ I had a math teacher in high school who refused to let us use calculators for addition, subtraction, multiplication or division.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

.All of these methods are commonly implemented in hardware floating-point units for performance reasons.^ So, basically, it only returns values in between here, which is just what we said, we are looking at that region of the graph between these points and that is all it is going to return.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ And, the reason it is going to be pi is if you just go back and look at your unit circle, right, that we have all come to know and love, this is zero radians, this is pi over two radians, and this is two pi over two radians.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Deriving approximations is nice and all, but there's really no point unless you do some sort of test to see how well they perform.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

.For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral.^ You get into middle/high school, learn about trigonometric functions, differential and integral calculus, exponentials, complex numbers (perhaps), etc.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ Another property of symmetrical functions is that, if you break them down into series expansions, odd functions will only have odd terms, and even functions only have even terms.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ From the definitions of the trigonometric functions, it is easy to see that h = a sin B = b sin A, so that a/sin A = b/sin B. By considering the altitude to another side, we see that this also means that a/sin A = b/sin B = c/sin C. These relations are called the Law of Sines.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

[10]
.Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples.^ Its use makes some computations simpler.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ So, if you look at some of these things and you are like man, it is not hitting me, just start with an angle and you will get good at your trig functions, you have to know this by the back of your hand.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ In the foregoing real addition theorems, take y equal to x, to yield the following double-angle formulae : .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.In fact, the sine, cosine and tangent of any integer multiple of π / 60 radians (3°) can be found exactly by hand.^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ You are taking the cosecant and rewriting it in terms of sine because personally, I am more confident and comfortable and familiar with working with sine, cosine and tangent.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ That is what this is equal to, the reason it is equal to that is because the tangent function is sine over cosine and if you remember back to the rainbow, I already showed you that cotangent is equal to one over tangent, so you have here is one over sine over cosine, so you have cosine over sine.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.Consider a right triangle where the two other angles are equal, and therefore are both π / 4 radians (45°).^ We say a triangle, that contains a right angle, is a right triangle.
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^ First of all, we say two angles are complementary if they add up to a right angle.
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^ But also, notice that if I put in the arcsin of one, notice that an equally valid angle is negative three pi over two.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π / 4 radians (45°) can then be found using the Pythagorean theorem:
c = \sqrt { a^2+b^2 } = \sqrt2\,.
Therefore:
\sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos \left(\pi / 4 \right) = \cos \left(45^\circ\right) = {1 \over \sqrt2},\,
	an \left(\pi / 4 \right) = 	an \left(45^\circ\right) = {{\sin \left(\pi / 4 \right)}\over{\cos \left(\pi / 4 \right)}} = {1 \over \sqrt2} \cdot {\sqrt2 \over 1} = {\sqrt2 \over \sqrt2} = 1. \,
.To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees).^ For an equilateral triangle, all three angles are equal, and equal 60° i.e.
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^ Thus all three sides of the triangle have equal length.
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^ [Note that an equilateral triangle has three vertex angles, all of which are 60°.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles.^ We say a triangle, that contains a right angle, is a right triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ I think you can convince yourself this is going to be pi over three because this is like 60 degrees and sine of 60 degrees is square root of three over two.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Cause, you can figure out sin, cos and tan, of 0 30 45 60 90, easily if you can remember the ratios in the 30-60-60 and the 45-45-90 triangles.
  • Is trig/algebra really that useful for calculus? | Pre-Medical Allopathic [ MD ] | Student Doctor Network 16 January 2010 10:48 UTC forums.studentdoctor.net [Source type: General]

For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:
\sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos \left(\pi / 3 \right) = \cos \left(60^\circ\right) = {1 \over 2}\,,
\cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin \left(\pi / 3 \right) = \sin \left(60^\circ\right) = {\sqrt3 \over 2}\,,
	an \left(\pi / 6 \right) = 	an \left(30^\circ\right) = \cot \left(\pi / 3 \right) = \cot \left(60^\circ\right) = {1 \over \sqrt3}\,.

Inverse functions

.The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function.^ This is not the only way of defining trigonometric functions; they can be defined as analytic functions of a complex variable z by power series, for example.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ But, in this section, we are going to focus on the inversed trig functions and how they kind of apply to Calculus.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ As the radius rotates, the trigonometric functions go through a regular cycle of variations with period 2π, the sine and cosine bounded by ±1, the tangent and cotangent going to ±∞.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

.Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective.^ The class containing the functions must also define a method named getNmbr(), which takes no parameters and returns the number of functions to be plotted.
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

^ We shall, however, revisit this problem subsequent to the study of inverse trigonometric function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and definitions .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.In the following, the functions on the left are defined by the equation on the right; these are not proved identities.^ So, what we are going to do when we define the arcsin function, is we are going to define it only to return values in a certain range and we are going to constrict that range to be right here.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ By substitution of the addition formulae into the right-hand side of the following, one may obtain these real product theorems : .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and definitions .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

The principal inverses are usually defined as:
 \begin{matrix} \mbox{for} & -\frac{\pi}{2} \le y \le \frac{\pi}{2}, & y = \arcsin x & \mbox{if} & x = \sin y \,;\\ \ \mbox{for} & 0 \le y \le \pi, & y = \arccos x & \mbox{if} & x = \cos y \,;\\ \ \mbox{for} & -\frac{\pi}{2} < y < \frac{\pi}{2}, & y = \arctan x & \mbox{if} & x = 	an y \,;\\ \ \mbox{for} & -\frac{\pi}{2} \le y \le \frac{\pi}{2}, y 
e 0, & y = \arccsc x & \mbox{if} & x = \csc y \,;\\ \ \mbox{for} & 0 \le y \le \pi, y 
e \frac{\pi}{2}, & y = \arcsec x & \mbox{if} & x = \sec y \,;\\ \ \mbox{for} & 0 < y < \pi, & y = \arccot x & \mbox{if} & x = \cot y \,. \end{matrix}
.For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc.^ So, this arcsin which is the same thing as this, it is just written differently, different notation, it is an inverse, literally, the inverse of the sine function.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ We shall, however, revisit this problem subsequent to the study of inverse trigonometric function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ If we write sin b sin c cos A = cos a - cos b cos c by rearranging the formula for cos a, and then substitute the formula for cos c in this, we can use cos 2 b = 1 - sin 2 b and then divide by sin b to find sin c cos A = cos a sin b - cos b sin a cos C. This formula can be used to find c after using the cot formula to find A in an SAS case, instead of using the Law of Sines directly.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

.When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions.^ So, this arcsin which is the same thing as this, it is just written differently, different notation, it is an inverse, literally, the inverse of the sine function.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Since the fifth order function also uses 4 multiplications and is considerably more accurate, you'd be better off using that.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ In this section of Advanced Calculus 2, we continue to cover calculus topics, and learn about the use of inverse trigonometric functions.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "arcsecond".
.Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series.^ I prefer to have sine over cosine like this.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

For example,
 \arcsin z = z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^7} {7} + \cdots\,.
.These functions may also be defined by proving that they are antiderivatives of other functions.^ This is not the only way of defining trigonometric functions; they can be defined as analytic functions of a complex variable z by power series, for example.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ The arcsin, the inverse sine, and by the way, by analogy, the inverse cosine, the inverse tangent, the inverse cotangent, all of these things, all of these trig functions you know, they all have an inverse.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

The arcsine, for example, can be written as the following integral:
 \arcsin z = \int_0^z \frac 1 {\sqrt{1 - x^2}}\,dx, \quad |z| < 1.
.Analogous formulas for the other functions can be found at Inverse trigonometric functions.^ These formulas for the functions of x + y and x - y are the basic formulas for deriving all the others.
  • Trigonometry 16 January 2010 10:48 UTC mysite.du.edu [Source type: Academic]

^ We shall, however, revisit this problem subsequent to the study of inverse trigonometric function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and definitions .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

Using the complex logarithm, one can generalize all these functions to complex arguments:
 \arcsin z = -i \log \left( i z + \sqrt{1 - z^2} \right), \,
 \arccos z = -i \log \left( z + \sqrt{z^2 - 1}\right), \,
 \arctan z = \frac{i}{2} \log\left(\frac{1-iz}{1+iz}\right).

Properties and applications

.The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results.^ So, what we have to do is find angle, but do not forget that because we are talking about the inverse trig function and we are talking about some angle we are getting back, the angle is going to be between negative pi over two and pi over two, by definition.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ This reduction is required as otherwise it would be difficult to estimate when two trigonometric functions together evaluates to minimum and maximum values.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ For any (x, y) in the Cartesian product of Complex by Complex, from the foregoing real addition theorems, with the replacement of y by i y and the use of the circular trigonometric functions , the following complex addition theorems are obvious .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},
or, equivalently,
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,
where R is the triangle's circumradius.
A Lissajous curve, a figure formed with a trigonometry-based function.
.It can be proven by dividing the triangle into two right ones and using the above definition of sine.^ The plotting surface is divided into the required number of equally sized plotting areas, and one function is plotted on cartesian coordinates in each area.
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

^ For an isosceles triangle, the two angles opposite the sides with equal length (i.e., the two angles whose bounding sides have one of the two sides of equal length, and the side with different length) are equal.
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^ This is not a unique choice:  The right-hand side of the first equation could have been multiplied by an arbitrary constant k.  Then, the right-hand sides of each of the other two equations divided by the same constant.  That is .
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.^ If a triangle has two sides of equal length and one side of different length, we call it an isosceles triangle.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ For an isosceles triangle, the two angles opposite the sides with equal length (i.e., the two angles whose bounding sides have one of the two sides of equal length, and the side with different length) are equal.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ By the way, the reason I used (1− z ) terms in the equation is because it's actually more of a cosine approximation than one for sines.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem:
c^2=a^2+b^2-2ab\cos C, \,
or equivalently,
\cos C=\frac{a^2+b^2-c^2}{2ab}.
.In this formula the angle at C is opposite to the side c.^ To be opposite to and delimit: The side of a triangle subtends the opposite angle.
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

^ For an isosceles triangle, the two angles opposite the sides with equal length (i.e., the two angles whose bounding sides have one of the two sides of equal length, and the side with different length) are equal.
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^ We know that the triangle we just created is isosceles, (two of its sides are radii), so the two angles opposite the radii have the same size, and add up to 120°: they are both 60°.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.^ This is the generalization of the Pythagorean theorem to non-right triangles.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ Another fact, of some use, is the Pythagorean theorem: H²=A²+O².
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^ The plotting surface is divided into the required number of equally sized plotting areas, and one function is plotted on cartesian coordinates in each area.
  • Understanding the Discrete Cosine Transform in Java — Developer.com 9 February 2010 13:38 UTC www.developer.com [Source type: FILTERED WITH BAYES]

.The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known.^ So, whatever angle you get has to be between zero and pi and that is going to be the angle that will give you the cosine of negative one.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ For an isosceles triangle, the two angles opposite the sides with equal length (i.e., the two angles whose bounding sides have one of the two sides of equal length, and the side with different length) are equal.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ This means that solving for the sine of an angle by the law of sines does not strictly determine the angle, normally.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

.It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ For an isosceles triangle, the two angles opposite the sides with equal length (i.e., the two angles whose bounding sides have one of the two sides of equal length, and the side with different length) are equal.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ [EXERCISE: Learn to use the sine law by explicitly writing out the equalities for the 45-45-90 triangle with hypotenuse , the 30-60-90 triangle with hypotenuse 2, and the equilateral triangle with all sides length 2.
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

Other useful properties

There is also a law of tangents:
\frac{a+b}{a-b} = \frac{	an[\frac{1}{2}(A+B)]}{	an[\frac{1}{2}(A-B)]}\,.

Sine and cosine of sums of angles

.Detailed, diagrammed construction proofs, by geometric construction, of formulas for the sine and cosine of the sum of two angles are available for download as a four-page PDF document at File:Sine Cos Proofs.pdf.^ What are the angle sum, difference, and/or doubling formulae?
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

^ The sine of five pi over six which is over here, pi over six, two pi over six, three pi over six, four pi over six, five pi over six and six pi over six would be a pi, so we know we are on track.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ The basic angle sum formulae are [A,B are arbitrary angles here]: .
  • Trigonometry: A Crash Review 12 January 2010 9:56 UTC www.zaimoni.com [Source type: FILTERED WITH BAYES]

Periodic functions

Animation of the additive synthesis of a square wave with an increasing number of harmonics
Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top); the basis functions have wavelengths λ/k (k = integer) shorter than the wavelength λ of the sawtooth itself (except for k = 1). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon
.The trigonometric functions are also important in physics.^ For this, we shall first recapitulate a bit of basics and important results and then emphasize: how can we conceive trigonometric ratio as a function?
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

.The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ Because any angle that we put in, no matter if we go on to infinity or negative infinity, any angle we put in is going to go into this function, we call the sine function and it is going to spit out a number, that is always going to be between this.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.The sine and cosine functions are one-dimensional projections of uniform circular motion.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ So, what we did was we wrote it in terms of the function, use sine and cosine because that helps you walk around the circle and plug in values.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.Trigonometric functions also prove to be useful in the study of general periodic functions.^ In this section of Advanced Calculus 2, we continue to cover calculus topics, and learn about the use of inverse trigonometric functions.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ The formal IEEE definitions are "(1) An instrument generally used to display the power distribution of an incoming signal as a function of frequency.
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

^ So, here, you have your sine function that you have studied for a long, long time and used in Cal One and Trig and other classes.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

.The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.^ Now such “disagreement” between the theory and the reality could be used to illustrate how all formulas originated simply as models to approximate future outcomes.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ The value of the rapid variation in air pressure due to a sound wave, measured in pascals , microbars , or dynes - all used interchangeable, but pascals is now the preferred term.
  • Pro Audio Reference S 3 February 2010 18:24 UTC www.rane.com [Source type: Reference]

[11]
.Under rather general conditions, a periodic function ƒ(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We just find an angle so that when you plug them into the tangent function, which the tangent function is the sine over the cosine function, you find an angle to give you the answer you want.
  • Inverse Trigonometric Functions - Advanced Calculus 2 Video – 5min.com 16 January 2010 10:48 UTC www.5min.com [Source type: Original source]

^ When cosine function is given as f(x) = Acos(kx), the period of cosine function is given by .
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

[12] Denoting the sine or cosine basis functions by φk, the expansion of the periodic function ƒ(t) takes the form:
f(t) = \sum _{k=1}^\infty c_k \varphi_k(t).
For example, the square wave can be written as the Fourier series
 f_	ext{square}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ( (2k-1)t \right )}\over(2k-1)}.
.In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation.^ While westerners imported exotic birds such as parrots and weavers, people in Eastern Europe introduced just a few game birds that were good for hunting.
  • History News Network 12 January 2010 9:56 UTC hnn.us [Source type: News]

The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

History

The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD). .The sine and cosine functions were discovered by Aryabhata (476–550) and studied by Varahamihira and Brahmagupta.^ Solution : The argument (input) to cosine function is sine function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Solution : The given function comprises of sine and cosine functions.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The MacLaurin's series for the hyperbolic sine and cosine may be obtained from their definitions and the series for the exponential function.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.The tangent function was discovered by Muhammad ibn Mūsā al-Khwārizmī (780–850), and the reciprocal functions of secant, cotangent and cosecant were discovered by Abū al-Wafā' Būzjānī (940–998).^ It follows that the secant, versed sine, and haversed sine are even, while the tangent, cotangent, and cosecant are odd.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ Each of the trigonometric functions will inherit this modulus of periodicity, except that the tangent and cotangent functions have a modulus of periodicity only half the size.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

^ The MacLaurin's series formulae for the hyperbolic arc sine, arc cosine, arc cotangent, arc secant, arc cosecant, arc versed sine, arc coversed sine, and arc haversed sine are not interesting.
  • Hyperbolic Trigonometric Functions 16 January 2010 10:48 UTC www.rism.com [Source type: Academic]

.All six trigonometric functions were then studied by Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.^ In particular, we shall come to know that some of these trigonometric functions are not defined for all values of angles.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ We shall, however, revisit this problem subsequent to the study of inverse trigonometric function.
  • Trigonometric functions 16 January 2010 10:48 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ All trig and precalc and high school calc is essentially the study of such functions.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

[citation needed]
Madhava of Sangamagrama (c. .1400) made early strides in the analysis of trigonometric functions in terms of infinite series.^ Another property of symmetrical functions is that, if you break them down into series expansions, odd functions will only have odd terms, and even functions only have even terms.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ The series is meant to have an infinite number of terms and when you truncate the series, you will lose some accuracy.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

[13]
The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician Albert Girard.
In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.[14]
Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.[1]
.A few functions were common historically, but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables [1]), the haversine (haversin(θ) = versin(θ) / 2 = sin2(θ/2)), the exsecant (exsec(θ) = sec(θ) − 1) and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1).^ Meanwhile, such problems seldom bring skills that students can even imagine a use for.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

^ Now, for the first time, a wider social group, including both aristocrats and commoners such as Pepys with fashionable aspirations, set the tone.
  • History News Network 12 January 2010 9:56 UTC hnn.us [Source type: News]

^ Now such “disagreement” between the theory and the reality could be used to illustrate how all formulas originated simply as models to approximate future outcomes.
  • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

.Many more relations between these functions are listed in the article about trigonometric identities.^ Related Flashcards dervitives of trig functions Trigonometry Functions Trig And Inverse Trig Derivatives Derivatives of Trig Functions Derivatives of Basic Functions trig stuff + more parent functions Functions, Domain and Range math Functions Functions 1.1-1.3 temperature Trig ratios converting from radians to degrees Fundemental Trig Identities Fundemental Trig Identities .
  • Basic Trig Functions - Degrees / Flashcards - Create Free Flashcards 16 January 2010 10:48 UTC www.proprofs.com [Source type: Academic]

^ From them you can see that there is a huge gap between the low-accuracy and high-accuracy functions of about a factor 60.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

^ I'd been wondering about that when I did my arctan article , but figured it would require too many terms to really be worth the effort.
  • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

Etymologically, the word sine derives from the Sanskrit word for half the chord, jya-ardha, abbreviated to jiva. This was transliterated in Arabic as jiba, written jb, vowels not being written in Arabic. .Next, this transliteration was mis-translated in the 12th century into Latin as sinus, under the mistaken impression that jb stood for the word jaib, which means "bosom" or "bay" or "fold" in Arabic, as does sinus in Latin.^ Translated into 20 languages, it was selling 50,000 copies a year a half century after it first appeared.
  • History News Network 12 January 2010 9:56 UTC hnn.us [Source type: News]

[15] Finally, English usage converted the Latin word sinus to sine.[16] .The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans — "cutting" — since the line cuts the circle.^ Although there's no evidence for an etymon of the word in French, the name presumably comes from Anglo-French coste (`rib') < Latin costa .
  • SBF Glossary: con to COXE 9 February 2010 13:38 UTC www.plexoft.com [Source type: FILTERED WITH BAYES]

^ Convent is an old word meaning `meeting, gathering, company,' from the Latin convenire, `convene.'
  • SBF Glossary: con to COXE 9 February 2010 13:38 UTC www.plexoft.com [Source type: FILTERED WITH BAYES]

See also

Notes

  1. ^ a b c See Boyer (1991).
  2. ^ See Maor (1998)
  3. ^ See Ahlfors, pages 43–44.
  4. ^ Abramowitz; Weisstein.
  5. ^ Stanley, Enumerative Combinatorics, Vol I., page 149
  6. ^ For a demonstration, see Euler's_formula#Using Taylor series
  7. ^ Needham, p. [ix "INSERT TITLE"]. ix. 
  8. ^ Kantabutra.
  9. ^ However, doing that while maintaining precision is nontrivial, and methods like Gal's accurate tables, Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used.
  10. ^ "R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM 23, 242 (1976).". http://doi.acm.org/10.1145/321941.321944. 
  11. ^ Stanley J Farlow (1993). Partial differential equations for scientists and engineers (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. ISBN 048667620X. http://books.google.com/books?id=DLUYeSb49eAC&pg=PA82. 
  12. ^ See for example, Gerald B Folland (2009). "Convergence and completeness". Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77 ff. ISBN 0821847902. http://books.google.com/books?id=idAomhpwI8MC&pg=PA77. 
  13. ^ J J O'Connor and E F Robertson. "Madhava of Sangamagrama". School of Mathematics and Statistics University of St Andrews, Scotland. http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html. Retrieved 2007-09-08. 
  14. ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. 
  15. ^ See Maor (1998), chapter 3, regarding the etymology.
  16. ^ "Clark University". http://www.clarku.edu/~djoyce/trig/. 

References

  • Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York. (1964). .ISBN 0-486-61272-4.
  • Lars Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, second edition, McGraw-Hill Book Company, New York, 1966.
  • Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition.^ As a guest, you can browse and view the various discussions in the forums, but can not create a new topic or reply to an existing one unless you are logged in.

    ^ Reply It’s easy to prove (in the mathematical sense) that any finite set of integers can be exactly modeled with at least one polynomial function.
    • Answer to the Friday Puzzle…. « Richard Wiseman's Blog 12 January 2010 9:56 UTC richardwiseman.wordpress.com [Source type: FILTERED WITH BAYES]

    ^ A new Russian history book for schools, approved by the Putin Government, glosses over Stalins Terror and other truths.
    • History News Network 12 January 2010 9:56 UTC hnn.us [Source type: News]

    (1991). ISBN 0-471-54397-7.
  • Gal, Shmuel and Bachelis, Boris. .An accurate elementary mathematical library for the IEEE floating point standard, ACM Transaction on Mathematical Software (1991).
  • Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed.^ I've also tested the standard floating-point sin() library function, the libnds sinLerp() and my own isin() function that you can find in arctan:sine .
    • Coranac » Another fast fixed-point sine approximation 3 February 2010 18:24 UTC www.coranac.com [Source type: FILTERED WITH BAYES]

    Penguin Books, London. (2000). .ISBN 0-691-00659-8.
  • Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans.^ You get into middle/high school, learn about trigonometric functions, differential and integral calculus, exponentials, complex numbers (perhaps), etc.
    • Why American consumers can't add - The Red Tape Chronicles - msnbc.com 12 January 2010 9:56 UTC redtape.msnbc.com [Source type: FILTERED WITH BAYES]

    Computers
    45 (3), 328–339 (1996).
  • Maor, Eli, Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
  • Needham, Tristan, "Preface"" to Visual Complex Analysis. .Oxford University Press, (1999).^ This notion is introduced in a textbook that was published in 1936 by Oxford University Press: Richard V. Southwell's An Introduction to the Theory of Elasticity for Engineers and Physicists .
    • SBF Glossary: con to COXE 9 February 2010 13:38 UTC www.plexoft.com [Source type: FILTERED WITH BAYES]

    ISBN 0-19-853446-9.
  • O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics archive. (1996).
  • O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics archive. (2000).
  • Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics archive. (2002).
  • Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.

External links


Citable sentences

Up to date as of December 14, 2010

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