Unit circle chart
The unit circle gives the sine, cosine, and tangent of the standard angles, in both degrees and radians. On a circle of radius 1, the cosine is the x-coordinate and the sine is the y-coordinate of the point at each angle.
Standard angles on the unit circle
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 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 |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −√3/3 |
| 180° | π | 0 | −1 | 0 |
| 210° | 7π/6 | −1/2 | −√3/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −√3/3 |
| 360° | 2π | 0 | 1 | 0 |
On the unit circle, a circle of radius 1 centered at the origin, each angle marks a point whose coordinates are the cosine (x) and sine (y). The tangent is the sine divided by the cosine, undefined where the cosine is zero. The values come from the special 30-60-90 and 45-45-90 right triangles.
Need the identities or circle formulas?
See the Trigonometric Identities Chart and the Circle Formula Chart.
Why the unit circle works
Set a circle of radius 1 at the origin and sweep a ray out at an angle. Where it meets the circle, the horizontal coordinate is the cosine of the angle and the vertical coordinate is the sine. Because the radius is 1, no scaling is needed, so the circle reads off trig values directly for any angle, all the way around.
Reading the special values
The recurring values, one half, root two over two, and root three over two, all come from two special right triangles. As the angle moves into the second, third, and fourth quadrants, the same magnitudes repeat with sign changes set by which coordinates are positive there. Cosine is negative on the left, sine is negative on the bottom.
FAQ
What is the unit circle?
A circle of radius 1 centered at the origin, used to define sine and cosine: the coordinates of the point at a given angle are its cosine and sine.
What is sin of 30 degrees?
One half. On the unit circle, the point at 30 degrees has a vertical coordinate of 0.5.
Why is tan 90 degrees undefined?
Tangent is sine over cosine, and at 90 degrees the cosine is zero. Division by zero is undefined, so the tangent does not exist there.