Radians

The circumference of a circle is $2\pi r$, where $r$ is the radius. Since the entire circumference subtends an angle of 360° at the center, we have the following relationship:

$2\pi \text{ radians} = 360°$

From this, we can derive the conversion factors:

$1 \text{ radian} = \frac{360°}{2\pi} \approx 57.3°$

$1° = \frac{2\pi}{360} = \frac{\pi}{180} \text{ radians} \approx 0.01745 \text{ radians}$

To convert an angle from degrees to radians, multiply the angle in degrees by $\pi/180$.

Convert the following angles from degrees to radians:

a) 60° b) 135° c) 270°

Solution:

a) $60° \times \frac{\pi}{180} = \frac{\pi}{3}$ radians b) $135° \times \frac{\pi}{180} = \frac{3\pi}{4}$ radians c) $270° \times \frac{\pi}{180} = \frac{3\pi}{2}$ radians

To convert an angle from radians to degrees, multiply the angle in radians by $180/\pi$.

Convert the following angles from radians to degrees:

a) $\pi/6$ radians b) $(2\pi/3)$ radians c) $(5\pi/4)$ radians

Solution:

a) $\frac{\pi}{6} \times \frac{180}{\pi} = 30°$ b) $\frac{2\pi}{3} \times \frac{180}{\pi} = 120°$ c) $\frac{5\pi}{4} \times \frac{180}{\pi} = 225°$

It's useful to know the trigonometric ratios of common angles in radians: