Trigonometry / Unit 1: Angles and Their Measure
1.2 Radian Measure
Notes
One radian is the measure of a central angle whose arc length equals the radius. A full turn is $2\pi$ radians.
- $180^\circ = \pi$ radians.
- Degrees to radians: multiply by $\displaystyle \frac{\pi}{180}$. Radians to degrees: multiply by $\displaystyle \frac{180}{\pi}$.
- Example: $\displaystyle 150^\circ \cdot \frac{\pi}{180} = \frac{5\pi}{6}$.
- Common values: $\displaystyle 30^\circ = \frac{\pi}{6}$, $\displaystyle 45^\circ = \frac{\pi}{4}$, $\displaystyle 60^\circ = \frac{\pi}{3}$, $\displaystyle 90^\circ = \frac{\pi}{2}$.
- One radian is about $57.3^\circ$.
Practice
Convert $150^\circ$ to radians.
$\displaystyle \frac{150}{180} = \frac{5}{6}$, so the angle is $\displaystyle \frac{5\pi}{6}$.
Convert $\displaystyle \frac{7\pi}{4}$ radians to degrees.
$\displaystyle \frac{7\pi}{4} \cdot \frac{180^\circ}{\pi} = 7 \cdot 45^\circ = 315^\circ$.
Approximately how many degrees is one radian?
$\displaystyle \frac{180}{\pi} \approx 57.3$.
Ready for more? Take the Unit 1 test.