From Right-Triangle Ratios to the Unit Circle (F 704)

Right-triangle trig only works for angles between 0 and 90 degrees, because that's all a triangle allows. The unit circle fixes that. Draw a circle of radius 1 centered at the origin, and let a point travel counterclockwise from (1, 0) by an angle d. That point's coordinates ARE (cos d, sin d), by definition. For angles under 90 degrees this matches the old triangle ratios exactly, since dropping a vertical line from the point to the x-axis builds a right triangle with hypotenuse 1. But now d can be 150 degrees, 210 degrees, even 700 degrees, and the point still lands somewhere with a definite x and y.

Because the circle has radius 1, every point on it satisfies x² + y² = 1, so cos²(d) + sin²(d) = 1 always. That's why sine and cosine never leave the interval from -1 to 1: they're coordinates on a circle of radius 1, and no coordinate on that circle can exceed 1 or drop below -1 in absolute value. The sign of each coordinate just tracks which quadrant the point sits in. In the third quadrant, both x and y are negative, so both sine and cosine are negative there, even though the angle itself is a perfectly ordinary positive number like 210 degrees.

This is the hinge that connects triangle trig to everything after it: radians, the sine wave, identities, and the Law of Sines and Law of Cosines all build on angles defined this way. Once sine and cosine are coordinates of a moving point, letting that angle grow past 360 degrees or trace out a full revolution is what generates the wave pattern you graph next.