Unit circle
The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. For any angle θ measured from the positive x-axis, the point where its ray meets the circle has coordinates (cos θ, sin θ).
The unit circle is the circle of radius 1 centered at the origin, with equation x² + y² = 1. Its power comes from how it defines the trigonometric functions: draw an angle θ in standard position (vertex at the origin, initial side along the positive x-axis), and the point where the terminal side crosses the circle is exactly (cos θ, sin θ). Cosine is the x-coordinate, sine is the y-coordinate, and tangent is sin θ / cos θ, the slope of the ray.
This definition extends right-triangle trigonometry (SOH CAH TOA) beyond acute angles. Because the point can lie in any quadrant, sine and cosine are defined for every angle — including angles past 90°, negative angles, and angles beyond a full revolution. The quadrant determines the signs: in quadrant II, for instance, sine is positive while cosine is negative.
Fluency with the unit circle means knowing the key angles. In degrees and radians: 0° (0), 30° (π/6), 45° (π/4), 60° (π/3), and 90° (π/2), whose sine values are 0, 1/2, √2/2, √3/2, and 1, with cosine running in reverse order. Every other quadrant repeats these reference values with sign changes. The circle also makes identities visible — the Pythagorean identity sin²θ + cos²θ = 1 is just the circle's equation.
The unit circle appears across standardized tests. The CLT tests unit circle trigonometry directly, the SAT uses it in radian and angle problems, and AMC 12 problems lean on it for trigonometric functions and identities. Memorizing the first-quadrant values and the sign pattern by quadrant covers most questions you will see.
Key takeaways
- The unit circle has radius 1, center at the origin, and equation x² + y² = 1.
- Any angle θ meets the circle at the point (cos θ, sin θ), defining sine and cosine for all angles.
- Key first-quadrant angles — 0°, 30°, 45°, 60°, 90° — have sine values 0, 1/2, √2/2, √3/2, 1.
- The Pythagorean identity sin²θ + cos²θ = 1 comes directly from the circle's equation.
- The CLT, SAT, and AMC 12 all test unit circle trigonometry.