The Unit Circle and Trigonometric Values from 0° to 360°

The unit circle is the circle of radius 1 centered at the origin, and for an angle θ measured from the positive x-axis, the point where the terminal side meets the circle has the coordinates (cos θ, sin θ). In brief: cosine is the x-coordinate and sine is the y-coordinate. At 120° the point is (−1/2, √3/2), so cos 120° = −1/2 and sin 120° = √3/2. The unit circle extends the trigonometric ratios from acute angles to all angles, with signs that depend on the quadrant.

The unit circle is the central picture of trigonometry. It shows why sine and cosine take values between −1 and 1, why the Pythagorean identity holds and how the values for angles from 0° to 360° follow from the values in the first quadrant. This guide explains the construction, the signs in the four quadrants, the reference angle method and gives a complete table of exact values at the standard angles.

How is the unit circle defined?

The unit circle has the equation x² + y² = 1. For an angle θ in standard position (vertex at the origin, initial side on the positive x-axis), the terminal side meets the circle at a point P(x, y). Then

• cos θ = x, sin θ = y,
• tan θ = y/x (undefined when x = 0), cot θ = x/y (undefined when y = 0).

For an acute angle, a right triangle with hypotenuse 1 and legs x and y recovers the definitions of the ratios in a right triangle. Since the point lies on the circle, x² + y² = 1 becomes cos²θ + sin²θ = 1 for every θ. Because x and y lie between −1 and 1, so do cosine and sine. The angle may be measured in degrees or radians.

What does the unit circle look like?

What are the signs in each quadrant?

A mnemonic is “All Students Take Calculus”: All positive in I, Sine positive in II, Tangent positive in III, Cosine positive in IV.

How do reference angles work?

The reference angle α of an angle θ is the acute angle between its terminal side and the x-axis. The trigonometric values of θ equal those of α, up to a sign given by the quadrant.

• Quadrant II: α = 180° − θ.
• Quadrant III: α = θ − 180°.
• Quadrant IV: α = 360° − θ.

Procedure: find the quadrant, compute α, take the values of α from the first quadrant and attach the sign.

Exact values from 0° to 360°

Worked examples

Example 1: 210°

The angle is in quadrant III, with reference angle 210° − 180° = 30°. Sine and cosine are negative there: sin 210° = −1/2, cos 210° = −√3/2 and tan 210° = (−1/2)/(−√3/2) = 1/√3 = √3/3.

Example 2: 7π/4

7π/4 = 315° is in quadrant IV with reference angle π/4. cos = +√2/2, sin = −√2/2 and tan = −1.

Example 3: a negative angle

−30° is coterminal with 330°: cos(−30°) = √3/2 and sin(−30°) = −1/2.

Example 4: from a point

The terminal side of θ passes through (−3, 4). The distance from the origin is r = 5. Then sin θ = 4/5, cos θ = −3/5, tan θ = −4/3. The angle is in quadrant II, about 126.87°.

Example 5: finding all angles with a given value

In [0°, 360°), sin θ = 1/2 gives θ = 30° and 150°, since sine is positive in quadrants I and II with reference angle 30°.

Where does the unit circle come from?

Greek and Indian astronomers used circles of a fixed radius (often 60 or 3,438 units) for their tables of chords and sines. The use of a circle with radius 1, which makes sine and cosine plain coordinates, became standard with Euler in the eighteenth century, who treated trigonometric functions as functions of a number: the length of the arc on the unit circle.

Where is it used today?

The unit circle describes anything that rotates or oscillates: a wheel, an alternating current, a sound wave. A point moving around the circle at constant speed has a horizontal position cos t and a vertical position sin t. Computer graphics use it to rotate objects, and navigation uses it to turn compass bearings into components.

Which mistakes are common?

Swapping sine and cosine

The point is (cos θ, sin θ): x first, y second. A reminder: “cosine comes before sine, as x comes before y”.

Wrong sign in the quadrant

Use the sign table, and use the reference angle only for the size.

Using the reference angle as the angle

The reference angle gives the size of the value, and the quadrant gives the sign. For 150°, the reference angle is 30°. Sine is positive in quadrant II, so sin 150° = sin 30° = 1/2, but cosine is negative there, so cos 150° = −cos 30° = −√3/2.

Assuming tan 90° exists

Tangent is undefined where cos θ = 0, at 90° and 270°.