#### Definition

Angles can be measured in units of degrees or radians. A complete revolution is defined as $360^\circ$ which is equal to $2\pi$ radians

$360^\circ = 2\pi \text{ radians.}$

From this, we can derive that

$1^{\circ} = \frac{\pi}{180} \text{ radians.}$ $1 \text{ radian} = \frac{180}{\pi}^{\circ}$

Sometimes a superscript c is used to denote radians instead of degrees, though it is conventional to assume that radians are used unless otherwise specified.

#### Worked Examples

###### Example 1

Convert $83^{\circ}$ to radians.

###### Solution

Recall that $1^{\circ} = \dfrac{\pi}{180} \text{ radians}$. So multiply $\dfrac{\pi}{180}$ by $83$:

\begin{align} 83^{\circ} &= 83 \times \frac{\pi}{180}\\ &\approx 1.449 \text{ radians (to 3 d.p.)} \end{align}

###### Example 2

Convert $3 \text{ radians}$ into degrees.

###### Solution

Using the definition, if $1 \text{ radian} = \dfrac{180}{\pi} \text{ degrees}$, then multiply by $3$ to find the angle in degrees.

$3 \times \frac{180}{\pi} \approx 172^{\circ} \text{ (to 3 sig.fig.)}$

#### Video Example

Prof. Robin Johnson shows how to convert $37^{\circ}$ to radians, and $1$ radian to degrees.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

#### More Support

You can get one-to-one support from Maths-Aid.