The unit circle & trigonometric functions
The idea
The unit circle frees trigonometry from triangles. Center a circle of radius 1 at the origin; an angle θ measured counterclockwise from the positive x-axis lands at a point whose coordinates ARE the trig values: (cos θ, sin θ). Suddenly sine and cosine make sense for any angle — 240°, negative angles, angles past a full turn — not just the acute angles a right triangle can hold. Radian measure belongs here too: an angle's radian measure is the arc length it cuts on the unit circle, with a full turn equal to 2π.
The working method is reference angles: every angle relates to an acute angle made with the x-axis, and the quadrant decides the signs — cosine follows the x-coordinate, positive on the right half, while sine follows the y-coordinate, positive on the top half. Memorize only the first-quadrant values for 30°, 45°, and 60° plus the axis points; everything else is a reference angle and a sign decision. The misconception to flush out is that sine or cosine can exceed 1: both are coordinates of a point glued to a radius-1 circle, so each lives forever inside [−1, 1].
Worked example
Find the exact values of cos 240° and sin 240°, convert 240° to radians, and verify that your values satisfy the Pythagorean identity.
- Locate the angle: 240° is 60° past 180°, so the terminal side lies in quadrant III, and the reference angle is 240° − 180° = 60°.
- Recall the first-quadrant values: cos 60° = 1/2 and sin 60° = √3/2. In quadrant III both coordinates are negative, so cos 240° = −1/2 and sin 240° = −√3/2.
- Convert to radians by multiplying by π/180: 240 × π/180 = 4π/3.
- Verify the identity: (−1/2)² + (−√3/2)² = 1/4 + 3/4 = 1. The point really does sit on the unit circle, exactly as the definition demands.
Answer. cos 240° = −1/2 and sin 240° = −√3/2, with 240° equal to 4π/3 radians.
Check your understanding
- Why does defining cosine and sine as coordinates extend them to angles that no right triangle could ever contain?
- How does an angle's quadrant determine the signs of its sine and cosine, and how could you rebuild that rule from the picture instead of memorizing it?
- What does the identity sin(θ + 360°) = sin θ mean geometrically on the unit circle?
- Why is radian measure considered the natural unit for angles once arc length enters the picture?
Build the foundations first
The unit circle & trigonometric functions builds on these concepts. If any feel shaky, start there.