The Unit Circle and Radian Problems

A right triangle only gives you sine and cosine for angles between 0 and 90 degrees. The unit circle fixes that. Draw a circle of radius 1 centered at the origin. Pick an angle, measured counterclockwise from the positive x-axis. Where the ray for that angle crosses the circle, call that point (x, y). By definition, x = cos(angle) and y = sin(angle). Because the radius is 1, there's no triangle to scale by, no hypotenuse to divide by — the coordinates just are the values.

Walking around the circle tells you the signs. Start at angle 0, the point (1, 0): cosine 1, sine 0. Move counterclockwise through the first quadrant and both coordinates stay positive. Cross 90 degrees into the second quadrant and x turns negative while y stays positive, so cosine goes negative and sine stays positive. Keep going: third quadrant, both negative; fourth quadrant, cosine positive again, sine negative. The angle 5 π over 6 (150 degrees) sits in the second quadrant, at (-√3/2, 1/2). For sin(3 π over 4), that's 135 degrees, in the second quadrant too, landing at (-√2/2, √2/2) — so sine is √2/2, read straight off the y-coordinate, no triangle needed.

This is why radians and degrees, and circle geometry generally, lean on the unit circle: it turns trig into coordinate geometry. Once you can place an angle in a quadrant, you know the signs of sine and cosine before computing anything, which catches sign errors fast and connects directly to arc length and radian measure on any circle.