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Radians
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.

External Resources
More Support
You can get one-to-one support from Maths-Aid .