In trigonometry, radian measure is an alternative to degree measure for angles. It is based on the length of the arc of a circle.
One radian is the angle at the centre of a circle which is subtended by an arc equal in length to the radius of the circle.
\[ 1 \text{ radian} = \frac{\text{Arc length}}{\text{Radius}} \]
Since the circumference of a circle is \( 2\pi r \) and a full revolution is \( 360^\circ \), \[ 360^\circ = 2\pi \text{ radians}. \] Therefore, \[ 1 \text{ radian} = \frac{180^\circ}{\pi}, \qquad 1^\circ = \frac{\pi}{180} \text{ radians}. \]
| Degrees (°) | Radians |
|---|---|
| 0° | 0 |
| 30° | \(\displaystyle \frac{\pi}{6}\) |
| 45° | \(\displaystyle \frac{\pi}{4}\) |
| 60° | \(\displaystyle \frac{\pi}{3}\) |
| 90° | \(\displaystyle \frac{\pi}{2}\) |
| 180° | \(\displaystyle \pi\) |
| 360° | \(\displaystyle 2\pi\) |
\[ 120^\circ \times \frac{\pi}{180^\circ} = \frac{120\pi}{180} = \frac{2\pi}{3}\ \text{radians}. \]
\[ \frac{3\pi}{4} \times \frac{180^\circ}{\pi} = \frac{3\times 180^\circ}{4} = 135^\circ. \]
One revolution = \( 360^\circ \). \[ 360^\circ \times \frac{\pi}{180^\circ} = 2\pi \ \text{radians}. \]