TrigonometryTrigonometryHigh school

The Unit Circle

Exact values at special angles and radian measure.

The unit circle has radius 1 and centre at the origin. For an angle $\theta$ measured counterclockwise from the positive $x$ -axis, the point where its terminal side meets the circle is $(\cos\theta, \sin\theta)$ . This definition extends sine and cosine to every angle, including obtuse, negative and very large ones.

Radians measure angles by arc length on the unit circle: a full turn is $2\pi$ , a half turn $\pi$ . Knowing the coordinates at $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ in the first quadrant, together with the signs in each quadrant, gives exact values everywhere else by symmetry.

Key terms

Unit-circle point: The point at angle t on the unit circle is (cos t, sin t).
$(\cos t, \sin t)$

Key ideas

$180^\circ = \pi$ radians; multiply degrees by $\frac{\pi}{180}$ to convert.
Signs: all positive in quadrant I, sine in II, tangent in III, cosine in IV.
Reference angle: the acute angle to the $x$ -axis, which fixes the absolute values.
$\sin^2\theta + \cos^2\theta = 1$ is the circle's equation $x^2 + y^2 = 1$ .

Worked example

Find the exact value of $\cos\frac{5\pi}{6}$ .

$\frac{5\pi}{6}$ is in quadrant II.
Reference angle: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$ .
$\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ .
Cosine is negative in quadrant II: $-\frac{\sqrt{3}}{2}$ .

Answer: $-\frac{\sqrt{3}}{2}$

Common mistakes

Swapping coordinates: the $x$ -coordinate is cosine, not sine.
Forgetting the quadrant sign.
Converting with $\frac{180}{\pi}$ when going from degrees to radians.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Convert $135^\circ$ to radians as a decimal.
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$\frac{3\pi}{4} \approx 2.356$
2.Find $\sin\frac{7\pi}{6}$ .
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$-\frac{1}{2}$
3.Find $\tan\frac{\pi}{4}$ .
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$1$

The Unit Circle

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