Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.1 The Pythagorean Identity
Notes
For every angle $\theta$, the point $(\cos\theta, \sin\theta)$ is on the unit circle, so $\sin^2\theta + \cos^2\theta = 1$.
- Rearranged: $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$.
- Given one of $\sin\theta$ or $\cos\theta$, find the other with the identity. The quadrant decides the sign.
- Example: $\displaystyle \sin\theta = \frac{4}{5}$ in quadrant II gives $\displaystyle \cos^2\theta = 1 - \frac{16}{25} = \frac{9}{25}$, so $\displaystyle \cos\theta = -\frac{3}{5}$.
- Related: $1 + \tan^2\theta = \sec^2\theta$.
Practice
If $\displaystyle \sin\theta = \frac{4}{5}$ and $\theta$ is in quadrant II, what is $\cos\theta$?
$\displaystyle \cos^2\theta = 1 - \frac{16}{25} = \frac{9}{25}$. Cosine is negative in quadrant II.
Which expression equals $1 - \sin^2 x$?
Rearrange $\sin^2 x + \cos^2 x = 1$.
If $\displaystyle \cos\theta = \frac{5}{13}$ and $\theta$ is in quadrant IV, what is $\sin\theta$?
$\displaystyle \sin^2\theta = 1 - \frac{25}{169} = \frac{144}{169}$. Sine is negative in quadrant IV.
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