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# Lesson 3:Trigonometric Identities: Wikis

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# Study guide

Up to date as of January 14, 2010

### From Wikiversity

Let us take a right angled triangle with hypotenuse length 1. If we mark one of the acute angles as θ, then using the definition of the sine ratio, we have

$\sin \theta = \cfrac{opposite}{hypotenuse}$

As the hypotenuse is 1,

$\sin \theta = \cfrac{opposite}{1} = opposite$

Repeating the same process using the definition of the cosine ratio, we have

$\cos \theta = \cfrac{adjacent}{hypotenuse} = \cfrac{adjacent}{1} = adjacent$

Since this is a right triangle, we can use the Pythagorean Theorem:

$\operatorname x^2 + y^2 = r^2$

$\operatorname \frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2}$

$\operatorname{cos}^2 \theta + \operatorname{sin}^2 \theta = 1$

This is the most fundamental identity in trigonometry.

$\operatorname \frac{x^2}{y^2} + \frac{y^2}{y^2} = \frac{r^2}{y^2}$

$\operatorname{cot}^2 x + 1 = \operatorname{csc}^2$

$\operatorname \frac{x^2}{x^2} + \frac{y^2}{x^2} = \frac{r^2}{x^2}$

$\operatorname 1 + \operatorname{tan}^2\theta = \operatorname{sec}^2\theta$

From this identity, if we divide through by squared cosine, we are left with:

$\cfrac{\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta}{\operatorname{cos}^2 \theta} = \cfrac{1}{\operatorname{cos}^2 \theta}$

$\operatorname{tan}^2 \theta + 1 = \operatorname{sec}^2\theta$

$\operatorname{sec}^2 \theta - \operatorname{tan}^2 \theta = 1$

If instead we divide the original identity by squared sine, we are left with:

$\cfrac{\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta}{\operatorname{sin}^2 \theta} = \cfrac{1}{\operatorname{sin}^2 \theta}$

$\operatorname{cot}^2 \theta + 1 = \operatorname{csc}^2 \theta$

$\operatorname{csc}^2 \theta - \operatorname{cot}^2 \theta = 1$

There are basically 3 main trigonometric identities. The proofs come directly from the definitions of these functions and the application of the Pythagorean theorem:

$\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta = 1$
$\operatorname{sec}^2 \theta - \operatorname{tan}^2 \theta = 1$
$\operatorname{csc}^2 \theta - \operatorname{cot}^2 \theta = 1$

Angle Sum-Difference Identities

$\sin(\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
$\cos(\alpha \pm \beta)=\cos \alpha\cos \beta\mp \sin \alpha \sin \beta$