Answer
$\color{green}{\theta = 210^\circ}$ .
$\color{blue}{\beta = 570^\circ}$
$\color{red}{\alpha = -150^\circ}$
Work Step by Step
$\color{green}{\theta = 210^\circ} $: Its initial side is ray $\vec{OA}$, and its terminal side is ray $\vec{OC}$, where $A=(1,0), O=(0,0),$ and $C=(\cos(210^\circ),\sin(210^\circ))$. $\theta$ is $\angle AOC$ measured in the counterclockwise direction. (See the figure.)
$\color{blue}{\beta = 570^\circ}$: Since $570^\circ = 210^\circ + 360^\circ$ ($\theta$ plus one counterclockwise revolution), then $\beta$ also has ray $\vec{OC}$ for its terminal side. $\theta$ and $\beta$ are therefore coterminal angles. ($\beta$ is not drawn in the figure.)
$\color{red}{\alpha = -150^\circ}$: This is $\angle AOC$ measured in the clockwise direction and is thus a negative angle that also has ray $\vec{OC}$ for its terminal side. (See the figure.)
