The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. The terminal side of any angle \theta in standard position crosses it at the point (\cos \theta, \sin \theta). So the x-coordinate is the cosine, the y-coordinate is the sine, and \tan \theta = \frac{y}{x}. For example, the point for 30° is (\frac{\sqrt{3}}{2}, \frac{1}{2}).

Before this lesson: You should know Reference Angles, Trig Values of Special Angles and Trig Functions of Any Angle.

Why Radius 1 Makes Trig Simple

For a point (x, y) on the terminal side, you know \sin \theta = \frac{y}{r} and \cos \theta = \frac{x}{r}. On the unit circle, r = 1. Dividing by 1 changes nothing, so \cos \theta = x and \sin \theta = y. The coordinates of the point are the cosine and sine.

The circle also shows a key fact. Every point satisfies x^2 + y^2 = 1, which is the equation of a circle with radius 1. So \cos^2 \theta + \sin^2 \theta = 1 for every angle.

Unit Circle Values

The first-quadrant points come from the special angles. Every other point uses the same numbers with the signs for its quadrant.

1. Find the reference angle of \theta.
2. Write the first-quadrant point for the reference angle.
3. Give each coordinate the sign for the quadrant: x is negative on the left, and y is negative below the x-axis.
4. Read \cos \theta from x and \sin \theta from y. Find \tan \theta = \frac{y}{x} if you need it.

Worked Examples

Example 1: Read sine and cosine from the circle

Find \sin 120^\circ and \cos 120^\circ using the unit circle.

180^\circ - 120^\circ = 60^\circ

(\frac{1}{2}, \frac{\sqrt{3}}{2})

(-\frac{1}{2}, \frac{\sqrt{3}}{2})

\sin 120^\circ = \frac{\sqrt{3}}{2}

\cos 120^\circ = -\frac{1}{2}

Answer: \sin 120^\circ = \frac{\sqrt{3}}{2} and \cos 120^\circ = -\frac{1}{2}

Example 2: An angle in radians

Find the point on the unit circle for \frac{7\pi}{4}, and give \sin \frac{7\pi}{4} and \cos \frac{7\pi}{4}.

2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4}

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})

Answer: \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2} and \sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}

Example 3: Tangent from the circle

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

(-\frac{\sqrt{3}}{2}, \frac{1}{2})

\tan \frac{5\pi}{6} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}

\frac{1}{2} \cdot (-\frac{2}{\sqrt{3}}) = -\frac{1}{\sqrt{3}}

-\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}

Answer: \tan \frac{5\pi}{6} = -\frac{\sqrt{3}}{3}

Example 4: Find the angles with a given sine

Find every angle \theta with 0 \le \theta \lt 2\pi and \sin \theta = -\frac{1}{2}.

Sine is the y-coordinate, so look for points with y = -\frac{1}{2}.

The reference angle with sine \frac{1}{2} is \frac{\pi}{6}.

Sine is negative in Quadrants III and IV.

\pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}

2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6}

Answer: \theta = \frac{7\pi}{6} or \theta = \frac{11\pi}{6}

Example 5: Word problem with a Ferris wheel

A Ferris wheel has a radius of 20 meters, and its center is 25 meters above the ground. A rider starts level with the center on the right side and turns 150° counterclockwise. How high is the rider now? Use height = 25 + 20 \sin \theta.

\sin 150^\circ = \frac{1}{2}

20 \cdot \frac{1}{2} = 10

25 + 10 = 35

Answer: The rider is 35 meters above the ground. The rider is also 20 \cos 150^\circ = 20 \cdot (-\frac{\sqrt{3}}{2}) = -10\sqrt{3} \approx -17.32 meters from the center horizontally, so about 17.32 meters to the left.

Common Mistakes

Related Methods

The pattern in the first quadrant

Going from 0° to 90°, the x-values shrink: 1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0. The y-values grow by the same list in reverse. Knowing this one list is enough to build the whole circle.

Symmetry

The circle is symmetric across both axes. The points for \theta, \pi - \theta, \pi + \theta and 2\pi - \theta have the same coordinates except for signs. This is the same idea as reference angles.

Toward trig graphs

As a point moves around the circle, its y-coordinate rises and falls between −1 and 1. Plotting that height against the angle gives the sine wave, which is the start of the next unit.

Practice Problems

1. Give the point on the unit circle for 60°.
2. Give the point on the unit circle for \frac{3\pi}{4}.
3. Find \sin 240^\circ.
4. Find \cos \frac{5\pi}{3}.
5. Find \tan \frac{4\pi}{3}.
6. Find \sin \pi and \cos \pi.
7. Find \sec \frac{2\pi}{3}.
8. Find every \theta with 0 \le \theta \lt 2\pi and \cos \theta = \frac{\sqrt{2}}{2}.
9. Which angle between 0 and 2\pi has the point (-\frac{\sqrt{3}}{2}, -\frac{1}{2})?
10. A bike wheel has a radius of 0.35 meter. A reflector starts at the 3 o’clock position and turns \frac{5\pi}{4} radians counterclockwise around the center. Find its position relative to the center, (0.35 \cos \frac{5\pi}{4}, 0.35 \sin \frac{5\pi}{4}), to the nearest hundredth.

\pi - \frac{3\pi}{4} = \frac{\pi}{4}

240^\circ - 180^\circ = 60^\circ

2\pi - \frac{5\pi}{3} = \frac{\pi}{3}

(-\frac{1}{2}, -\frac{\sqrt{3}}{2})

\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}

\sec \frac{2\pi}{3} = \frac{1}{-\frac{1}{2}} = -2

2\pi - \frac{\pi}{4} = \frac{7\pi}{4}

\pi + \frac{\pi}{6} = \frac{7\pi}{6}

(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})

0.35 \cdot (-\frac{\sqrt{2}}{2}) \approx -0.247

-0.247 \approx -0.25

FAQ

What is the unit circle?

It is a circle with radius 1 centered at the origin. For any angle \theta, the point where its terminal side meets the circle is (\cos \theta, \sin \theta).

How do you memorize the unit circle?

Learn the three first-quadrant points for 30°, 45° and 60°. Every other point uses the same numbers with the signs for its quadrant.

Is sine x or y on the unit circle?

Sine is the y-coordinate, and cosine is the x-coordinate. Alphabetical order helps: cosine before sine, x before y.

How do you find tangent on the unit circle?

Divide the y-coordinate by the x-coordinate: \tan \theta = \frac{\sin \theta}{\cos \theta}. It is undefined where x = 0, at \frac{\pi}{2} and \frac{3\pi}{2}.

Summary

• The unit circle has radius 1 and center (0, 0).
• The point for angle \theta is (\cos \theta, \sin \theta).
• \tan \theta = \frac{y}{x}, and \cos^2 \theta + \sin^2 \theta = 1.
• First-quadrant points come from 30°, 45° and 60°; other points use reference angles and signs.
• Most sine or cosine values appear at two angles between 0 and 2\pi.

Previous lesson: Reference Angles

Next lesson: Graphs of Sine and Cosine

Back to the Trigonometry Index