Periodic Phenomena and Angle Measure

What "periodic" actually means

A function $f$ is periodic with period $k$ when $f(x+k)=f(x)$ for every $x$ in the domain, and $k$ is the smallest such positive number. Two consequences the exam leans on:

• $f(x+nk)=f(x)$ for every integer $n$ — so one period of information determines the whole function;
• the period is read between two consecutive matching states: maximum to next maximum, or any point to the next point with the same value and the same direction of motion.

Radian measure

The radian measure of an angle is the ratio $\theta=\dfrac{s}{r}$: arc length over radius. It is a pure number, which is why $s=r\theta$ works with no conversion factor — and why it fails instantly if $\theta$ is in degrees.

Conversion: multiply by $\dfrac{\pi}{180}$ to go from degrees to radians. The benchmark angles $\dfrac{\pi}{6},\dfrac{\pi}{4},\dfrac{\pi}{3},\dfrac{\pi}{2}$ must be automatic.

Co-terminal angles differ by a multiple of $2\pi$: on the circle, $\theta$ and $\theta+2\pi n$ point at the same spot.