Reference Angle
A reference angle is the acute angle formed between the terminal side of an angle in standard position and the nearest x-axis.
Definition
A reference angle is the acute angle, between $0^\circ$ and $90^\circ$, formed between the terminal side of an angle in standard position and the nearest $x$-axis. In each quadrant: Q1, $\text{ref} = \theta$; Q2, $\text{ref} = 180^\circ - \theta$; Q3, $\text{ref} = \theta - 180^\circ$; Q4, $\text{ref} = 360^\circ - \theta$. More precisely, for $\theta$ in $[0, 2\pi)$, the reference angle is $\min(\theta \bmod \pi, \pi - (\theta \bmod \pi))$, and this concept is implicit in the reduction formulas $\sin(\pi - \theta) = \sin(\theta)$, $\sin(\pi + \theta) = -\sin(\theta)$, and similar identities.
Example
The reference angle for $150^\circ$ is $30^\circ$ (since $150^\circ$ is $30^\circ$ from the $180^\circ$ line), and the reference angle for $210^\circ$ is also $30^\circ$. For $\theta = 240^\circ$ (Q3), the reference angle $= 240^\circ - 180^\circ = 60^\circ$; since Q3 has $\sin < 0$ and $\cos < 0$, $\sin(240^\circ) = -\sin(60^\circ) = -\sqrt{3}/2$. The reduction formulas express trig values of any angle in terms of its reference angle: $\sin(5\pi/6) = \sin(\pi - \pi/6) = \sin(\pi/6) = 1/2$.
Key Insight
Reference angles let you use your knowledge of $30^\circ$, $45^\circ$, and $60^\circ$ angles to find trig values for angles in any quadrant, the reference angle gives the size and the quadrant gives the sign; this collapses a $360^\circ$ problem into a $90^\circ$ one. Reference angles correspond to the fundamental domain of the dihedral group acting on the circle: the eight symmetries of the circle (reflections and rotations by multiples of $\pi/2$) generate all reduction formulas from the single interval $[0, \pi/2]$.