Skill: Identify reference angles

A reference angle is the positive acute angle between an angle’s terminal side and the $x$-axis. To find it:

1. Rewrite the angle between $(0^\circ)$ and $(360^\circ)$ by adding or subtracting $(360^\circ)$ if needed.
2. Identify the quadrant of the terminal side.
3. Use the quadrant rule:
   • Quadrant I: $\alpha=\theta$
   • Quadrant II: $\alpha=180^\circ-\theta$
   • Quadrant III: $\alpha=\theta-180^\circ$
   • Quadrant IV: $\alpha=360^\circ-\theta$
4. If the terminal side lies on an axis, the reference angle is $(0^\circ)$.

Example: Find the reference angle for $\theta=-225^\circ$.

First, locate the angle within one revolution. Since the angle is negative, add $(360^\circ)$:

$$-225^\circ+360^\circ=135^\circ$$

So, $(-225^\circ)$ is coterminal with $(135^\circ)$. They have the same terminal side.

Next, identify the quadrant. Since

$$90^\circ<135^\circ<180^\circ,$$

the terminal side is in Quadrant II.

For an angle in Quadrant II, subtract the angle from $(180^\circ)$:

$$\alpha=180^\circ-135^\circ=45^\circ$$

Therefore, the reference angle is

$$\boxed{45^\circ}$$

The terminal side is $(45^\circ)$ above the negative $x$-axis, so the reference angle is positive and acute.