# Circular functions

## Radians and degrees

$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$

## Exact values



## $\sin$ and $\cos$ graphs

$$f(x)=a \sin(bx-c)+d$$
$$f(x)=a \cos(bx-c)+d$$

where
$a$ is the amplitude
$b$ is the $x$-dilation
$c$ is the $y$-shift

Period is ${2 \pi} \over b$
Domain is $\mathbb{R}$
Range is $[-b+c, b+c]$;

Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.

**Mean / equilibrium:** line that the graph oscillates around ($y=d$)

## Solving trig equations

1. Solve domain for $n\theta$
2. Find solutions for $n\theta$
3. Divide solutions by $n$

$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$