24 min read

The Unit Circle

The unit circle defines sin and cos for any angle, not just angles inside a right-angled triangle.

Beyond the Triangle

Draw a circle of radius 1 centred at the origin. Starting from the point (1, 0), measure an angle θ going anti-clockwise. The point on the circle at that angle has coordinates (cos(θ), sin(θ)). This definition works for ANY angle — including angles greater than 90 °, which do not fit inside a right-angled triangle.

Unit circle

Angle 30°

cos(30°) ≈ 0.866 · sin(30°) ≈ 0.500 · point ≈ (0.866, 0.500)

Change a value and watch:

Drag the angle around the circle. cos(θ) is the horizontal position, sin(θ) is the vertical position.

At θ = 90 °, the point is (0, 1), so cos(90 °) = 0 and sin(90 °) = 1 — matching what you already know.

Example 1 — checking against known values

Confirm that the unit circle gives the same value as SOH CAH TOA for θ = 30 °. 1 On the unit circle, sin(30 °) is the y -coordinate of the point at 30 °. 2 That value is 0.5, matching the special-angle value from the last lesson. The unit circle and SOH CAH TOA agree for all angles between 0 ° and 90 °.

Practice

On the unit circle, what are the coordinates of the point at θ = 180 °?