Algebra 2 / Unit 8: Trigonometric Functions
8.6 Basic Trigonometric Identities
Notes
An identity is an equation true for every allowed value.
- Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.
- Quotient identities: $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta}$.
- Reciprocal identities: $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.
- To simplify, rewrite everything in sine and cosine.
- Example: $\displaystyle \frac{1 - \cos^2\theta}{\sin\theta} = \frac{\sin^2\theta}{\sin\theta} = \sin\theta$.
Practice
If $\cos\theta = 0.6$, find $\sin^2\theta$.
$\sin^2\theta = 1 - 0.36 = 0.64$.
Simplify $\displaystyle \frac{\sin\theta}{\tan\theta}$.
$\displaystyle \sin\theta \cdot \frac{\cos\theta}{\sin\theta} = \cos\theta$.
Evaluate $\displaystyle \csc\frac{\pi}{6}$.
$\displaystyle \csc = \frac{1}{\sin}$, and $\displaystyle \sin\frac{\pi}{6} = \frac{1}{2}$.
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