Trigonometry Calculator
Compute sine, cosine, and tangent for any angle — with a live unit circle.
0.7854 rad
On the unit circle, an angle’s cosine is the x-coordinate and its sine the y-coordinate; tangent is sin ÷ cos. For 30°, sin = 0.5, cos = √3 ÷ 2 ≈ 0.866, and tan = 1 ÷ √3 ≈ 0.577. The identity sin²θ + cos²θ = 1 always holds.
Sine, cosine, and tangent on the unit circle
Draw a ray from the centre of a circle of radius 1 at some angle. Where it meets the circle, the x-coordinate is the angle’s cosine and the y-coordinate is its sine. Tangent is the ratio of the two, sin ÷ cos — the slope of the ray.
the Pythagorean identity, and tangent as a ratio
Worked example
Find sin, cos, and tan of 30°:
- 1 Convert to radians if needed. 30° × π ÷ 180 = π/6 ≈ 0.5236 rad — the native functions work in radians.
- 2 Read the coordinates. sin 30° = 0.5 (the y-coordinate) and cos 30° = √3 ÷ 2 ≈ 0.8660 (the x-coordinate).
- 3 Divide for tangent. tan 30° = 0.5 ÷ 0.8660 = 1 ÷ √3 ≈ 0.5774. Check: sin² + cos² = 0.25 + 0.75 = 1. ✓
Common angle values
The exact values worth memorising; decimals rounded to four places.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.8660 | 0.5774 |
| 45° | 0.7071 | 0.7071 | 1 |
| 60° | 0.8660 | 0.5 | 1.7321 |
| 90° | 1 | 0 | undefined |
Degrees, radians, and the tan gap
A full circle is 360° or 2π radians. Convert with degrees × π ÷ 180, or radians × 180 ÷ π. Toggle the unit on the tool to match your problem.
Tangent is undefined at 90° and 270°. There cosine is zero, and dividing sine by zero has no value — the tangent graph shoots off to ±∞ at those points.
Signs follow the quadrant. Sine is positive above the x-axis, cosine positive to the right of the y-axis; tangent is positive where sine and cosine share a sign (the first and third quadrants).