What the unit circle is
The unit circle is the circle of radius 1 centered at the origin. Start on the positive x-axis and turn counterclockwise through an angle θ: the point where you stop has coordinates (cos θ, sin θ). That single idea defines sine and cosine for every angle, not just the acute angles of a right triangle, and it's why the unit circle is the backbone of trigonometry. Use this page to learn the exact values, to check homework, or to see why sine is negative in the third quadrant.
How the values are found
radians = degrees × π ÷ 180 cos²θ + sin²θ = 1
The exact values come from two special right triangles. A 45°-45°-90° triangle with hypotenuse 1 has legs √2/2. A 30°-60°-90° triangle with hypotenuse 1 has legs 1/2 and √3/2. Every one of the 16 standard angles has one of these triangles as its reference triangle, so only the signs change from quadrant to quadrant. Tangent is undefined at 90° and 270°, where x = 0.
Worked example: 150°
150° is in Quadrant II, where x is negative and y is positive. Its reference angle is 180° − 150° = 30°, and cos 30° = √3/2, sin 30° = 1/2. So the point is (−√3/2, 1/2): cos 150° = −√3/2 ≈ −0.8660, sin 150° = 1/2, and tan 150° = (1/2) ÷ (−√3/2) = −√3/3 ≈ −0.5774. In radians, 150° = 5π/6.
The 16 standard angles
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Where |
|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 | positive x-axis |
| 30° | π/6 | √3/2 | 1/2 | √3/3 | Quadrant I |
| 45° | π/4 | √2/2 | √2/2 | 1 | Quadrant I |
| 60° | π/3 | 1/2 | √3/2 | √3 | Quadrant I |
| 90° | π/2 | 0 | 1 | undefined | positive y-axis |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 | Quadrant II |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 | Quadrant II |
| 150° | 5π/6 | −√3/2 | 1/2 | −√3/3 | Quadrant II |
| 180° | π | −1 | 0 | 0 | negative x-axis |
| 210° | 7π/6 | −√3/2 | −1/2 | √3/3 | Quadrant III |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 | Quadrant III |
| 240° | 4π/3 | −1/2 | −√3/2 | √3 | Quadrant III |
| 270° | 3π/2 | 0 | −1 | undefined | negative y-axis |
| 300° | 5π/3 | 1/2 | −√3/2 | −√3 | Quadrant IV |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 | Quadrant IV |
| 330° | 11π/6 | √3/2 | −1/2 | −√3/3 | Quadrant IV |
Signs by quadrant and reference angles
Cosine takes the sign of x and sine takes the sign of y. In Quadrant I all three are positive; in II only sine is; in III only tangent is; in IV only cosine is. The reference angle is the acute angle between the terminal side and the x-axis: θ in Quadrant I, 180° − θ in II, θ − 180° in III and 360° − θ in IV. Sine and cosine of any angle have the same size as those of its reference angle.
Limits
- Exact values are given for the 16 standard angles (multiples of 30° and 45°), and for any coterminal angle such as 405° or −π/4. Other angles get decimals to 10 significant figures.
- Radians typed as decimals, like 0.7854, are treated as approximate. Type π/4 or pi/4 to get exact answers.