13A

Radian measure of angle size

The definition of a radian, converting between degrees and radians, exact trig values of special angles in radians, and using the calculator in radian mode.

What a radian is

One radian is the angle at the centre of a circle that cuts off an arc equal in length to the radius. A full circle has circumference $2\pi r$, so a revolution is $2\pi$ radians.

Conversion

$$\pi \text{ radians} = 180^\circ \qquad 1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.3^\circ$$

Radians on the unit circle

On a circle of radius 1, the angle in radians equals the arc length from (1, 0) to the end of the blue ray.

Try this

Slide to $\theta = \pi \approx 3.14$. What angle in degrees is that?

Example 1●

Convert $150^\circ$ to radians and $\dfrac{3\pi}{4}$ to degrees.

$150 \times \dfrac{\pi}{180} = \dfrac{5\pi}{6}$

Multiply degrees by π/180.

$\dfrac{3 \times 180^\circ}{4} = 135^\circ$

Replace π by 180°.

Special angles

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

Exact values work exactly as before: use the special triangles and ASTC. For example, $\cos\dfrac{2\pi}{3} = -\cos\dfrac{\pi}{3} = -\dfrac12$, since $\dfrac{2\pi}{3}$ is in quadrant 2.

Flashcards

$pi$ radians

$180^\circ$

1 radian

$\approx 57.3^\circ$

Degrees to radians

Multiply by $\dfrac{\pi}{180}$.

$rac{pi}{6}, rac{pi}{4}, rac{pi}{3}$

$30^\circ, 45^\circ, 60^\circ$

Practice

Convert $150^\circ$ to exact radians (type like 5pi/6).
Convert $\dfrac{3\pi}{4}$ to degrees.
How many degrees is $1$ radian, to 1 decimal place?
Find $\sin\dfrac{\pi}{6}$.
Find $\cos\dfrac{2\pi}{3}$.
Find $\tan\dfrac{5\pi}{4}$.
Find $\sin 1.2$ (radians), correct to 3 decimal places.
$270^\circ$ in radians is:
Find $\cos\pi$.