Sine, Cosine, Tangent
The calculator finds the sine, cosine, tangent and cotangent of an angle in degrees or radians, converts degrees to radians, shows the quadrant, signs and reduction formulas. Exact values with roots.
The trigonometry calculator finds the sine, cosine, tangent and cotangent of an angle given in degrees or radians, and converts degrees to radians and back. For angles that are multiples of 15° it gives exact values with roots (for example, sin 210° = −1/2, tan 75° = 2 + √3), finds the quadrant and the signs of the functions, and shows the reduction formulas.
Degrees and radians
A full turn is 360° or 2π radians, so 180° = π rad. This gives the conversion formulas:
αrad = α° · π / 180°, α° = αrad · 180° / π
Example 1. 210° = 210 · π / 180 = 7π/6 ≈ 3.665 rad.
Example 2. 2π/3 = 2π/3 · 180°/π = 120°. 1 rad ≈ 57°17′45″.
Values for the main angles
| α | 0° | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) |
|---|---|---|---|---|---|
| sin α | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos α | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan α | 0 | √3/3 | 1 | √3 | — |
| cot α | — | √3 | 1 | √3/3 | 0 |
For 15° and 75°: sin 15° = cos 75° = (√6 − √2)/4, tan 15° = cot 75° = 2 − √3.
Signs by quadrant
| Quadrant | Angles | sin | cos | tan, cot |
|---|---|---|---|---|
| I | 0° … 90° | + | + | + |
| II | 90° … 180° | + | − | − |
| III | 180° … 270° | − | − | + |
| IV | 270° … 360° | − | + | − |
Reduction formulas
Any angle can be reduced to an acute reference angle β: first drop the full 360° turns, then write the angle as 180° − β, 180° + β or 360° − β. The function stays the same, and the sign is taken from the quadrant of the original angle.
Example 3. sin 210° = sin(180° + 30°) = −sin 30° = −1/2 (quadrant III, sine is negative).
Example 4. cos 765° = cos(45° + 2 · 360°) = cos 45° = √2/2.
The tangent is undefined when cos α = 0 (α = 90° + 180°k), the cotangent when sin α = 0 (α = 180°k).
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