Topic 1 of 8

The unit circle and radians

Step 14 · Trigonometry and precalculus topics

Measure angles in radians, find the cosine and sine of an angle as the coordinates of a point on the unit circle, and use reference angles in every quadrant.

Radians

A radian is the angle that cuts off an arc as long as the radius. A full turn has an arc of 2π radii, so 360° is 2π radians.

180^{\circ} = \pi \text{ radians}

Degrees	30°	45°	60°	90°	180°	360°
Radians	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	π	2π

Example: Converting

Convert 150° to radians, and $\frac{3\pi}{4}$ to degrees.

$150 \times \frac{\pi}{180} = \frac{5\pi}{6}$ .
$\frac{3\pi}{4} \times \frac{180}{\pi} = 135$ .

Answer $\frac{5\pi}{6}$ and 135°

Example: Arc length

A circle has a radius of 10 cm. How long is the arc cut off by an angle of 1.5 radians?

In radians, arc length = radius × angle: s = rθ.
10 × 1.5 = 15.

Answer15 cm

Points on the unit circle

The unit circle has radius 1 and center (0, 0). Turn an angle θ counterclockwise from the positive x-axis: the point you reach is (cos θ, sin θ).

The unit circle. Turning 30° (π/6) from the x-axis reaches (√3/2, 1/2), so cos 30° = √3/2 and sin 30° = 1/2.

θ	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
cos θ	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
sin θ	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1

Example: Reading the circle

Find cos π and $\sin\frac{3\pi}{2}$ .

π is half a turn: the point (−1, 0).
$\frac{3\pi}{2}$ is three quarters of a turn: the point (0, −1).

Answer $\cos\pi = -1$ and $\sin\frac{3\pi}{2} = -1$

Reference angles

A reference angle is the sharp angle between the end of the turn and the x-axis. It gives the size of the cosine and sine; the quadrant gives their signs.

Quadrant	cos	sin	tan
I	+	+	+
II	−	+	−
III	−	−	+
IV	+	−	−

Example: Quadrant II

Find cos 150° and sin 150°.

The reference angle is 180 − 150 = 30°.
In quadrant II, cosine is negative and sine is positive.

Answer $\cos 150^{\circ} = -\frac{\sqrt{3}}{2}$ and $\sin 150^{\circ} = \frac{1}{2}$

Example: Quadrant IV

Find $\sin\frac{7\pi}{4}$ .

$\frac{7\pi}{4}$ is in quadrant IV, with reference angle $\frac{\pi}{4}$ .
Sine is negative there.

Answer $-\frac{\sqrt{2}}{2}$

Example: Going round again

Which angle from 0° to 360° ends in the same place as 480°?

Take away a full turn: 480 − 360 = 120.

Answer120°

Try it yourself

Work each one on paper first, then open the answer to check.

Convert 240° to radians.
Show answer
$\frac{4\pi}{3}$
Convert $\frac{\pi}{5}$ to degrees.
Show answer
36°
A circle has radius 4. How long is the arc for an angle of 2.5 radians?
Show answer
10
What is $\sin\frac{\pi}{2}$ ?
Show answer
1
Find cos 210°.
Show answer
$-\frac{\sqrt{3}}{2}$
Find tan 135°.
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−1
Which angle from 0 to 2π ends in the same place as $-\frac{\pi}{3}$ ?
Show answer
$\frac{5\pi}{3}$