Precalculus / Unit 5: Trigonometric Identities and Equations
5.1 Fundamental Identities
Notes
An identity is an equation true for every allowed value of the variable.
- Reciprocal: $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.
- Quotient: $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$, $\displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta}$.
- Pythagorean: $\sin^2\theta + \cos^2\theta = 1$, $1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$.
- Example: if $\displaystyle \sin\theta = \frac{3}{5}$ in quadrant II, then $\displaystyle \cos\theta = -\sqrt{1 - \frac{9}{25}} = -\frac{4}{5}$.
Practice
If $\displaystyle \sin\theta = \frac{3}{5}$ and $\theta$ is in quadrant II, what is $\tan\theta$?
$\displaystyle \cos\theta = -\frac{4}{5}$, so $\displaystyle \tan\theta = \frac{3/5}{-4/5} = -\frac{3}{4}$.
Which expression equals $\sec^2 x - \tan^2 x$?
Rearranging $1 + \tan^2 x = \sec^2 x$ gives $\sec^2 x - \tan^2 x = 1$.
Simplify $\displaystyle \frac{1 - \cos^2 x}{\sin x}$.
$1 - \cos^2 x = \sin^2 x$, so the result is $\displaystyle \frac{\sin^2 x}{\sin x} = \sin x$.
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