Trigonometry / Unit 1: Angles and Their Measure
1.3 Coterminal and Reference Angles
Notes
Coterminal angles share the same terminal side. Add or subtract full turns: $360^\circ$ (or $2\pi$ radians).
- $60^\circ$, $420^\circ$ and $-300^\circ$ are coterminal.
- The reference angle is the acute angle between the terminal side and the $x$-axis. It is always between $0^\circ$ and $90^\circ$.
- Quadrant II: $180^\circ - \theta$. Quadrant III: $\theta - 180^\circ$. Quadrant IV: $360^\circ - \theta$.
- Example: for $\displaystyle \frac{11\pi}{6}$ (Quadrant IV) the reference angle is $\displaystyle 2\pi - \frac{11\pi}{6} = \frac{\pi}{6}$.
Practice
What is the smallest positive angle coterminal with $-50^\circ$?
$-50^\circ + 360^\circ = 310^\circ$.
What is the reference angle for $225^\circ$?
$225^\circ$ is in Quadrant III, so subtract $180^\circ$.
What is the reference angle for $\displaystyle \frac{11\pi}{6}$?
It is in Quadrant IV: $\displaystyle 2\pi - \frac{11\pi}{6} = \frac{\pi}{6}$.
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