Start with a circle of radius one
Measure the angle from the positive horizontal axis, with positive rotation counterclockwise. The corresponding point on the circle has horizontal coordinate cos θ and vertical coordinate sin θ.
Radians connect angle and arc length
On a unit circle, an angle’s radian measure equals the length of the arc it spans. One complete turn is 2π radians or 360°, so a quarter turn is π/2 radians or 90°.
A mistake to avoid
Sine and cosine are signed coordinates, not always positive lengths. Their signs change by quadrant. Do not mix a degree input with a calculation that expects radians.
Worked example
Read the point at a quarter turn
At θ = 90° = π/2 radians, the unit-circle point is (0, 1). Therefore cos θ = 0 and sin θ = 1. At a half turn, θ = π and the point is (−1, 0), giving cos θ = −1 and sin θ = 0.