The Unit Circle: A Visual Explanation of Sine and Cosine
See why cosine is the x-coordinate, sine is the y-coordinate and signs change by quadrant on the unit circle.
Quick answer
On a circle of radius 1 centered at the origin, the point reached by rotating through angle θ has coordinates (cosθ, sinθ). Cosine is the horizontal coordinate, sine is the vertical coordinate, and the signs follow the point's quadrant.
Key relationships
Build sine and cosine from coordinates
Start at the positive x-axis and rotate a radius through angle θ. The endpoint's horizontal projection is x = cosθ and its vertical projection is y = sinθ. This turns trigonometric ratios into coordinates that work beyond acute triangles.
Because every endpoint lies one unit from the origin, x² + y² = 1. Substituting x = cosθ and y = sinθ produces the Pythagorean identity.
Use quadrants instead of memorizing signs
In quadrant I, x and y are positive, so cosine and sine are positive. In quadrant II, x is negative and y positive. Quadrant III makes both negative. Quadrant IV makes cosine positive and sine negative.
The geometry explains the sign pattern directly. There is no need to store it as an unrelated rule if you can picture the coordinate axes.
Reuse reference angles
Points with the same reference angle form matching right triangles in different quadrants. Their coordinate magnitudes repeat while their signs change. For example, 30°, 150°, 210° and 330° all use the same 30° reference triangle.
This symmetry is also why sine and cosine repeat periodically. One complete revolution returns to the same point, so adding 2π radians does not change either coordinate.
What to remember
- A unit-circle point has coordinates (cosθ, sinθ).
- The radius is 1, so cos²θ + sin²θ = 1.
- Quadrants determine the signs of sine and cosine.
- Reference angles reuse the same coordinate magnitudes around the circle.
Frequently asked questions
Why is cosine the x-coordinate?
In the right triangle formed by dropping a vertical from the unit-circle point, the adjacent side is x and the hypotenuse is 1, so cosθ = x/1 = x.
Why is sine sometimes negative?
Sine is the y-coordinate. Points below the x-axis have negative y-values, so their sine values are negative.
Why use radians on the unit circle?
On a unit circle, an angle in radians equals the arc length it subtends, which makes calculus relationships especially natural.
Watch sine and cosine emerge from the unit circle
Play the complete lesson free—drawn and narrated from start to finish, with no account needed.
