Skill: Determine coterminal angles

Coterminal angles have the same initial side and terminal side in standard position. Their measures differ by a whole number of full rotations.

Use the rule

$$\theta_{\text{coterminal}}=\theta+360^\circ k,$$

where $k$ is any integer. A positive $k$ adds full rotations, while a negative $k$ subtracts full rotations. In radians, use $2\pi k$ instead of $360^\circ k$.

Example: Find two coterminal angles for $-45^\circ$.

Start with the given angle:

$$\theta=-45^\circ$$

To find a positive coterminal angle, add one full rotation:

$$-45^\circ+360^\circ=315^\circ$$

So, $315^\circ$ is coterminal with $-45^\circ$.

To find another coterminal angle, subtract one full rotation:

$$-45^\circ-360^\circ=-405^\circ$$

Therefore, two coterminal angles are

$$\boxed{315^\circ \text{ and } -405^\circ}.$$

These angles are not numerically equal to $-45^\circ$, but they have the same terminal side because each differs from $-45^\circ$ by a whole number of full rotations.