The Unit Circle Explained: Sine and Cosine for Any Angle

Q: Why is the radius of the unit circle 1?

With a radius of 1, the hypotenuse of the triangle is 1, so the sine and cosine are simply the coordinates of the point, with no division needed.

Q: How do I find sin or cos of an angle over 90°?

Find the reference angle (the acute angle to the x-axis), take its sine or cosine, then give it the sign for the quadrant the angle is in.

How putting a right-angled triangle in a circle of radius 1 extends sine and cosine to any angle, the signs in each quadrant, reference angles, and the exact values to remember.

3 min read Supports 4 lessons

SOH CAH TOA only works inside a right-angled triangle, so it only covers angles between $0°$ and $90°$ . But an angle can be $150°$ or $300°$ or more. The unit circle gives every angle a sine and a cosine.

The idea

Draw a circle of radius $1$ centred on the origin. Start at $(1, 0)$ and turn anticlockwise through an angle $\theta$ . Drop a line down to the $x$ -axis and you have a right-angled triangle whose hypotenuse is $1$ .

With a hypotenuse of $1$ , $\cos\theta = \frac{\text{adj}}{1}$ is just the horizontal leg, and $\sin\theta$ is the vertical leg. So the point's coordinates are the cosine and sine. Drag the point:

Unit circle definition

P(\theta) = (\cos\theta,\ \sin\theta) \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Key idea

Now the definition works for any angle. Past $90°$ , the point is still somewhere on the circle, with coordinates that may be negative.

Signs in each quadrant

Cosine is the $x$ -coordinate and sine the $y$ -coordinate, so their signs follow the quadrant. Tangent is positive when the two have the same sign.

Where each is positive

\text{QI: all} \qquad \text{QII: } \sin \qquad \text{QIII: } \tan \qquad \text{QIV: } \cos

Often remembered as CAST, read anticlockwise from the bottom right.

Reference angles

Every angle has a reference angle: the acute angle it makes with the $x$ -axis. The sine and cosine of any angle equal those of its reference angle, give or take a sign.

$\sin 150^\circ = \;?$
150° is in quadrant II.

1/4

Exact values

The 30-60-90 triangle gives $\sin 30° = \frac12$ , $\cos 30° = \frac{\sqrt3}{2}$ .

Tip

A pattern for the first quadrant: $\sin$ of $0°, 30°, 45°, 60°, 90°$ is $\frac{\sqrt0}{2}, \frac{\sqrt1}{2}, \frac{\sqrt2}{2}, \frac{\sqrt3}{2}, \frac{\sqrt4}{2}$ . Cosine runs the same list backwards.

What it leads to

Because every point on the unit circle is distance $1$ from the origin, Pythagoras gives $\cos^2\theta + \sin^2\theta = 1$ for every angle. And tracking the height of the point as it goes round and round traces out the sine wave.

Common questions

Why is the radius of the unit circle 1?: With a radius of $1$ , the hypotenuse of the triangle is $1$ , so the sine and cosine are simply the coordinates of the point, with no division needed.
How do I find sin or cos of an angle over 90°?: Find the reference angle (the acute angle to the $x$ -axis), take its sine or cosine, then give it the sign for the quadrant the angle is in.

Practise unit circle

The The Unit Circle lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why is the radius of the unit circle 1?

With a radius of $1$ , the hypotenuse of the triangle is $1$ , so the sine and cosine are simply the coordinates of the point, with no division needed.

How do I find sin or cos of an angle over 90°?

Find the reference angle (the acute angle to the $x$ -axis), take its sine or cosine, then give it the sign for the quadrant the angle is in.