Unit Circle Values
Complete reference for sine, cosine, and tangent values at key angles (0°, 30°, 45°, 60°, 90° and beyond).
Essential trig reference table.
The Unit Circle
The unit circle is a circle with radius 1 centered at the origin.
For any angle θ, the point on the unit circle is (cos θ, sin θ).
Key Angle Values (First Quadrant)
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
All Four Quadrants
| Degrees | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
| 120° | √3/2 | -1/2 | -√3 |
| 135° | √2/2 | -√2/2 | -1 |
| 150° | 1/2 | -√3/2 | -√3/3 |
| 180° | 0 | -1 | 0 |
| 210° | -1/2 | -√3/2 | √3/3 |
| 225° | -√2/2 | -√2/2 | 1 |
| 240° | -√3/2 | -1/2 | √3 |
| 270° | -1 | 0 | undefined |
| 300° | -√3/2 | 1/2 | -√3 |
| 315° | -√2/2 | √2/2 | -1 |
| 330° | -1/2 | √3/2 | -√3/3 |
| 360° | 0 | 1 | 0 |
Sign Rules by Quadrant
Remember which functions are positive in each quadrant with the mnemonic: All Students Take Calculus
- Quadrant I (0° to 90°): All are positive
- Quadrant II (90° to 180°): Sin is positive
- Quadrant III (180° to 270°): Tan is positive
- Quadrant IV (270° to 360°): Cos is positive
Example 1
Find sin(150°)
150° is in Quadrant II. The reference angle is 180° - 150° = 30°.
sin is positive in Quadrant II.
sin(150°) = sin(30°) = 1/2
Example 2
Find cos(225°)
225° is in Quadrant III. The reference angle is 225° - 180° = 45°.
cos is negative in Quadrant III.
cos(225°) = -cos(45°) = -√2/2
When to Use It
Use the unit circle reference when:
- You need exact trig values without a calculator
- Solving trig equations and finding all solutions
- Determining the sign of a trig function in a given quadrant
- Converting between degrees and radians
Key Notes
- Key (cos θ, sin θ) pairs on the unit circle: 0° (1, 0); 30° (√3/2, 1/2); 45° (√2/2, √2/2); 60° (1/2, √3/2); 90° (0, 1). The pattern in the first quadrant: sin values read 0, 1, √2, √3, 2 (divided by 2) as angle increases — this symmetry makes memorization systematic.
- Extending to all quadrants with ASTC: All (0–90°): all trig positive. Sine (90–180°): only sin positive. Tangent (180–270°): only tan positive. Cosine (270–360°): only cos positive. Mnemonic: "All Students Take Calculus." Signs flip systematically from the first-quadrant values.
- Reference angles: For any angle θ, the reference angle is the acute angle between the terminal side and the x-axis. The trig function magnitudes equal those of the reference angle; the ASTC rule determines the sign. θ = 150°: reference = 30°; sin 150° = +sin 30° = 0.5.
- Period and symmetry: sin and cos repeat every 360° (2π radians); tan repeats every 180° (π). sin is an odd function: sin(−θ) = −sin θ. cos is even: cos(−θ) = cos θ. These symmetry properties halve the table you need to memorize.
- Applications: Unit circle values are used constantly in physics (resolving vectors into components), engineering (phase angles in AC circuits), calculus (derivatives and integrals of trig functions use exact values), signal processing (computing Fourier coefficients at standard frequencies), and navigation (compass bearing to vector components).