From triangle to circle

The right-triangle definition of sin and cos works perfectly — but only for angles between 0° and 90°. The unit circle extends both to any angle at all, and reveals what sin and cos really are.

Setting up the unit circle

Draw a circle of radius 1 centered at the origin. For any angle θ, draw the radius that makes that angle with the positive x-axis. The point where the radius meets the circle is:

(cos θ, sin θ)

That's the definition. Not a formula derived from something else — the coordinates are cos and tan.

The right triangle lives inside

For 0° < θ < 90°, drop a perpendicular from P down to the x-axis. You get a right triangle with:

• Hypotenuse = 1 (the radius)
• Adjacent leg along the x-axis = cos θ (the x-coordinate)
• Opposite leg going up = sin θ (the y-coordinate)

The triangle definition and the circle definition give the same values — the unit circle is just the triangle with hypotenuse forced to 1.

What the unit circle adds

At θ = 90°, the radius is straight up: P = (0, 1), so cos 90° = 0 and sin 90° = 1. At θ = 180°, P = (−1, 0): cos 180° = −1, sin 180° = 0.

The triangle couldn't reach these angles. The unit circle extends sin and cos to all angles — and makes their signs visible, not just their magnitudes. That's what the next lesson is about.