3.1 Trig Functions of Any Angle

Notes

For an angle in standard position, let $(x, y)$ be any point on the terminal side (not the origin) and let $\displaystyle r = \sqrt{x^2 + y^2}$. Then the trig functions work for any angle, not just acute ones.

• $\displaystyle \sin\theta = \frac{y}{r}$, $\displaystyle \cos\theta = \frac{x}{r}$, $\displaystyle \tan\theta = \frac{y}{x}$.
• $\displaystyle \csc\theta = \frac{r}{y}$, $\displaystyle \sec\theta = \frac{r}{x}$, $\displaystyle \cot\theta = \frac{x}{y}$.
• $r$ is always positive.
• Example: for $(-3, 4)$, $r = 5$, so $\displaystyle \cos\theta = -\frac{3}{5}$.