Trigonometric and Polar Functions

Notes

Periodic functions and the unit circle

A periodic function repeats: $f(x + P) = f(x)$ for period $P$.

• An angle in radians is arc length divided by radius. $180° = \pi$ radians.
• On the unit circle, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$, and $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$.

Sinusoidal functions

$f(\theta) = a\sin(b(\theta + c)) + d$

• Amplitude $|a|$, period $\displaystyle \frac{2\pi}{|b|}$, midline $y = d$, phase shift $-c$.
• Modeling: amplitude is half of (max − min); midline is the average of max and min.

Tangent and inverse trig

• $\tan\theta$ has period $\pi$ and vertical asymptotes where $\cos\theta = 0$, like $\displaystyle \theta = \frac{\pi}{2}$.
• $\sin^{-1}$ has outputs in $\displaystyle \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, $\cos^{-1}$ in $[0, \pi]$, $\tan^{-1}$ in $\displaystyle \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Identities and other trig functions

• Pythagorean: $\sin^2\theta + \cos^2\theta = 1$.
• Sum: $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$. Double angle: $\sin 2\theta = 2\sin\theta\cos\theta$.
• $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.

Polar coordinates

A point $(r, \theta)$ is $r$ units from the origin at angle $\theta$. Convert with $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$.

• In a polar graph $r = f(\theta)$, when $r$ is positive and increasing, the point is moving away from the origin.