Angles in Standard Position

Standard position gives every angle a fair starting line: its vertex sits at the origin, and its initial side lies along the positive x-axis. From there the angle opens up like a rotating arm. Positive angles sweep counterclockwise, negative angles sweep clockwise, and wherever the arm stops is the terminal ray. The size of the angle just tells you how far that arm rotated to get there.

Because a full turn is 360 degrees, rotating 360 degrees more (or less) brings the arm right back to where it started, pointing the same direction. That is why 30 degrees and 390 degrees look identical on a diagram: 390 is just 30 plus one extra full spin. Angles like these, which share a terminal ray, are called coterminal. You can find a coterminal angle by adding or subtracting any multiple of 360 degrees, and every angle has infinitely many of them. What actually matters for the direction of the ray is the angle's remainder after dividing by 360, not the raw number of degrees.

This idea sets up everything else in the unit. Once an angle is pinned to a terminal ray, that ray crosses the unit circle at a specific point, and the coordinates of that point become the building blocks for sine and cosine. Angles beyond 360 degrees, or negative angles winding the other way, all collapse onto one of these points, which is exactly why trigonometric functions repeat forever. Standard position is the bridge between rotation and the wave you get when you unwrap the circle.