Coterminal angles

Rotating the ray $OA$ anticlockwise through one full turn produces an angle of $360^\circ$ but lands the terminal side back exactly where it started. Nothing stops the rotation from continuing — anticlockwise rotations beyond $360^\circ$, or clockwise rotations giving negative angles, all keep landing the terminal side on the same ray, just after a different number of turns.

Angles that share the same initial side and the same terminal side — even though reached by different amounts of rotation — are called coterminal angles. If $\alpha$ and $\beta$ are coterminal, their measures differ by a whole number of complete rotations:

Worked check. $417^\circ$ and $-303^\circ$ are coterminal, since $417^\circ - (-303^\circ) = 720^\circ = 2(360^\circ)$ — an exact integer multiple of $360^\circ$. …