Unit Circle
The 16 standard angles in degrees and radians, with exact cos, sin, and tan.
The point sits at (cos θ, sin θ) = (√2/2, √2/2). Pick an angle above to highlight it on the circle.
The unit circle is a circle of radius 1 centred at the origin. A point on it at angle θ (measured counter-clockwise from the positive x-axis) has coordinates (cos θ, sin θ). So the x-coordinate is the cosine and the y-coordinate is the sine; tan θ is their ratio, sin θ ÷ cos θ.
What the unit circle shows
Because the radius is 1, the horizontal and vertical legs of the right triangle formed by any radius are exactly cos θ and sin θ. Sweeping the angle θ around the circle traces every possible pair of sine and cosine values, which is why the unit circle is the reference picture for all of trigonometry. The 16 standard angles — multiples of 30° and 45° — have exact values built from √2, √3, and simple fractions.
Reading the table
Each row gives an angle in both degrees and radians alongside its exact cosine (the x-coordinate), sine (the y-coordinate), and tangent. Tangent is undefined at 90° and 270°, where cos θ = 0 and the ratio sin θ ÷ cos θ divides by zero.
The 16 standard angles
Exact values. cos θ is the x-coordinate, sin θ the y-coordinate; tan θ = sin θ ÷ cos θ.
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | −√3/2 | 1/2 | −√3/3 |
| 180° | π | −1 | 0 | 0 |
| 210° | 7π/6 | −√3/2 | −1/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −1/2 | −√3/2 | √3 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 300° | 5π/3 | 1/2 | −√3/2 | −√3 |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 |
| 330° | 11π/6 | √3/2 | −1/2 | −√3/3 |
| 360° | 2π | 1 | 0 | 0 |
Why cos is x and sin is y
Drop a radius from the centre to a point on the circle and complete the right triangle back to the x-axis. The hypotenuse is the radius, length 1. By definition cosine is adjacent ÷ hypotenuse and sine is opposite ÷ hypotenuse; with a hypotenuse of 1 those become simply the horizontal leg (the x-coordinate) and the vertical leg (the y-coordinate). That is why the point is written (cos θ, sin θ).
How the signs follow the quadrant
The coordinate signs of the point decide the signs of cosine and sine. In quadrant I (0°–90°) both x and y are positive, so cos and sin are positive. In quadrant II (90°–180°) x is negative and y positive, so cos is negative and sin positive. In quadrant III (180°–270°) both are negative. In quadrant IV (270°–360°) x is positive and y negative. For example 210° gives (−√3/2, −1/2) and 300° gives (1/2, −√3/2).
Degrees, radians, and arc length
Radians measure the angle by the arc length it cuts on a circle of radius 1: a full turn is the whole circumference, 2π, so 360° = 2π radians and 180° = π. To convert, multiply degrees by π ÷ 180, or multiply radians by 180 ÷ π. That is why 30° = π/6 and 45° = π/4.