The Unit Circle

You’ll be able to

Interpret cosine and sine as the x- and y-coordinates on the unit circle
Evaluate trig values at the common special angles
Locate the quadrantal-angle points on the unit circle

Cosine and sine are coordinates

The unit circle is the circle of radius 1 centered at the origin. For an angle θ measured counterclockwise from the positive x-axis, the point where its ray meets the circle has coordinates (cos θ, sin θ). So cosine is the x-coordinate and sine is the y-coordinate. Because the radius is 1, these values always fall between −1 and 1.

The special angles

A handful of angles appear constantly, and their exact values are worth knowing. At π/6 (30°), cos = √3/2 and sin = 1/2. At π/4 (45°), cos = sin = √2/2. At π/3 (60°), cos = 1/2 and sin = √3/2. Notice the symmetry: as the angle grows from 30° to 60°, cosine shrinks while sine grows, and at 45° they meet.

Point on the unit circle

(x, y) = (cos θ, sin θ)

Quadrantal angles: θ = 0 → (1, 0); θ = π/2 → (0, 1); θ = π → (−1, 0); θ = 3π/2 → (0, −1).

Worked example

Find cos(π/3) and sin(π/3) using the unit circle.

1.π/3 is 60°, one of the special angles.
2.The x-coordinate at 60° is cos(π/3) = 1/2.
3.The y-coordinate at 60° is sin(π/3) = √3/2.
4.Check: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1, confirming the point lies on the unit circle.

Answer: cos(π/3) = 1/2 and sin(π/3) = √3/2. Cosine is the x-coordinate and sine is the y-coordinate of the 60° point.

Checkpoint

What is cos(π/3)?

Tip

Remember the pattern for sine at 0, 30, 45, 60, 90°: √0/2, √1/2, √2/2, √3/2, √4/2 — that is 0, 1/2, √2/2, √3/2, 1. Cosine runs the same list backward.

Checkpoint

The point on the unit circle at angle π/2 has coordinates:

On the exam

Sign by quadrant: in QII cosine is negative and sine positive; in QIII both are negative; in QIV cosine is positive and sine negative. Combine a reference angle with the quadrant’s sign to get any exact value.

Answer the 2 checkpoints as you read.