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Interactive unit circle with exact trig values, quadrant filtering, and ASTC sign rule.
The unit circle is a circle of radius 1 centered at the origin. For any angle θ, the coordinates of the point on the circle are (cos θ, sin θ), making it the foundation of trigonometric functions.
| Deg | Rad | sin | cos | tan | (x, y) |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | (√3/2, 1/2) |
| 45° | π/4 | √2/2 | √2/2 | 1 | (√2/2, √2/2) |
| 60° | π/3 | √3/2 | 1/2 | √3 | (1/2, √3/2) |
| 90° | π/2 | 1 | 0 | undefined | (0, 1) |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 | (-1/2, √3/2) |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 | (-√2/2, √2/2) |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 | (-√3/2, 1/2) |
| 180° | π | 0 | -1 | 0 | (-1, 0) |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 | (-√3/2, -1/2) |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 | (-√2/2, -√2/2) |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 | (-1/2, -√3/2) |
| 270° | 3π/2 | -1 | 0 | undefined | (0, -1) |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 | (1/2, -√3/2) |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 | (√2/2, -√2/2) |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 | (√3/2, -1/2) |
| 360° | 2π | 0 | 1 | 0 | (1, 0) |
Mnemonic: All Students Take Calculus
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
sin(−θ) = −sin θ
cos(−θ) = cos θ
sin(90° − θ) = cos θ
Formula
For angle θ on the unit circle: x = cos θ, y = sin θ, tan θ = y/xθ = The angle measured counterclockwise from the positive x-axis
(x, y) = Coordinates of the point on the unit circle
r = Radius = 1 (by definition of unit circle)
Worked Example
Find trig values for 150° (5π/6)
Did you know? The unit circle concept dates back to ancient Greek mathematician Hipparchus (190-120 BC), who created the first trigonometric table. The modern unit circle with radian measure was formalized by Leonhard Euler in the 18th century.
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