Unit Circle Calculator – Exact Trigonometric Values
Calculate exact sine, cosine, and tangent values for any angle on the unit circle.
Unit Circle: Key Angles and Exact Values
The unit circle has radius 1 centered at the origin. For any angle θ, the point on the circle is (cos θ, sin θ). The unit circle is fundamental to trigonometry, defining sine, cosine, and tangent for all angles beyond the original right-triangle definitions.
| Angle (°) | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
Key identities: sin²θ + cos²θ = 1 (Pythagorean identity). Sine is an odd function (sin(−θ) = −sin θ); cosine is even (cos(−θ) = cos θ). The period of both sin and cos is 2π, meaning values repeat every 360°. Tangent has period π (180°).
How do I memorize unit circle values?
For angles 0°, 30°, 45°, 60°, 90°: sine values are √0/2, √1/2, √2/2, √3/2, √4/2 (simplified: 0, 1/2, √2/2, √3/2, 1). Cosine is the reverse sequence.
Why is the unit circle important?
It extends trigonometry to all real numbers and angles, not just acute angles in triangles. It underlies Fourier transforms, complex numbers (Euler's formula: e^(iθ) = cos θ + i sin θ), and all oscillatory physics.