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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

Quiz: Lesson_3:Trigonometric_Identities_quiz


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