Radian Measure

Video lesson4 worked examples

Theory

Previously when measuring angles, we have always used degrees.

Radians are another unit of measure for angles.

One radian is the angle made at the centre of a circle by an arc which has length equal to the radius of the circle. Since a radian is defined by the arc of a circle, we can think of radian as 'distance' travelled on a unit circle.

\text{Radian} = \frac{\text{distance travelled}}{\text{radius}}

Conversion Rules

To convert Degrees to Radians: Multiply by $\frac{\pi}{180}$
To convert Radians to Degrees: Multiply by $\frac{180}{\pi}$

Degrees	Radians
$180^\circ$	$\pi$
$360^\circ$	$2\pi$
$30^\circ$	$\frac{\pi}{6}$
$45^\circ$	$\frac{\pi}{4}$
$60^\circ$	$\frac{\pi}{3}$
$90^\circ$	$\frac{\pi}{2}$
$270^\circ$	$\frac{3\pi}{2}$
$225^\circ$	$\frac{5\pi}{4}$

⚠️ Common Examiner Traps

Never mix the two in one line: mixing degrees and radians within a single line of working is a frequent and expensive error. Decide which the question is in and stay there.
Answer in the units asked for: working in degrees throughout when radians were wanted is a common mistake. If the domain is given as $0 \leq x \leq 2\pi$ , the answer must be in radians.
Set your calculator to the right mode: a correct method in the wrong mode gives a wrong answer with no way back. Check it before paper 2 and every time the units change.
Practise this deliberately: radian measure and exact values are a known weak spot across the course — not a minor detail.

Worked examples

Example 1

Convert $61^\circ$ to radians.

\begin{aligned} \text{Radians} &= 61 \times \frac{\pi}{180} \\ &= \frac{61\pi}{180} \end{aligned}

Example 2

Convert $65^\circ$ to radians.

\begin{aligned} \text{Radians} &= 65 \times \frac{\pi}{180} \\ &= \frac{65\pi}{180} \\ &= \frac{13\pi}{36} \end{aligned}

Example 3

Convert $\frac{7\pi}{4}$ to degrees.

\begin{aligned} \text{Degrees} &= \frac{7\pi}{4} \times \frac{180}{\pi} \\ &= 7 \times 45 \\ &= 315^\circ \end{aligned}

Example 4

Convert $\frac{5\pi}{6}$ to degrees.

\begin{aligned} \text{Degrees} &= \frac{5\pi}{6} \times \frac{180}{\pi} \\ &= 5 \times 30 \\ &= 150^\circ \end{aligned}