Radians and conversion

One radian is the central angle subtending an arc equal in length to the radius. More generally, \(\theta=\dfrac lr\) radians. Since a complete circumference is \(2\pi r\), a full turn is \(2\pi\) radians: \(180^\circ=\pi\text{ radians}\).

Example

\(150^\circ=150\times\dfrac\pi{180}=\dfrac{5\pi}{6}\text{ radians}\). Conversely, \(\dfrac{7\pi}{12}\text{ radians}=105^\circ\). An arc of length 12 cm on a circle of radius 8 cm subtends \(\dfrac{12}{8}=1.5\) radians, approximately \(85.9^\circ\).

A radian angle has no degree symbol. A value of 1.5 radians is not \(1.5^\circ\). Label the intended unit and use the matching calculator mode for trigonometric calculations.