Sine and Cosine on the Unit Circle

Sine and cosine live on a circle of radius one. Here's the unit circle: radius is exactly one, centered at the origin. Now I'll mark an angle theta measured counterclockwise from the positive x-axis.

The point on the circle at angle theta has coordinates cosine theta, sine theta.

Let me show you with a concrete example: angle of thirty degrees. At thirty degrees, cosine is the horizontal distance from the center. And sine is the vertical distance from the center.

• cos(30°) = √3/2 ≈ 0.866
• sin(30°) = 1/2 = 0.5

Notice: cosine squared plus sine squared always equals one.

This comes directly from the Pythagorean theorem on the unit circle. As theta increases, the point rotates counterclockwise around the circle. At ninety degrees, cosine is zero and sine is one.

• Radius is 1, so x² + y² = 1
• cos(90°) = 0, sin(90°) = 1

Quick check: what is cosine of zero degrees? Sine and cosine are the x and y coordinates of any point on the unit circle.