Trigonometry Calculator
Get all six trig functions of an angle in degrees, radians or gradians, with exact values like √3/2 for special angles, the reference angle, quadrant and a unit-circle drawing. Switch to inverse mode to find the angle from a value, with every solution between 0° and 360°.
Numbers or expressions with π: pi/6, 2pi/3, 3π/4
Decimals or expressions: 1/2, √3/2, sqrt(2)/2, -1
How to use the Trigonometry Calculator
- Choose the mode: functions of an angle, or inverse functions to find an angle.
- Enter the angle (e.g. 30 or 5pi/6) and pick degrees, radians or gradians.
- Read the six values: exact forms appear for special angles.
- For inverse functions, pick arcsin, arccos, … and enter a value such as √3/2.
- Check the unit circle and the steps, or copy all the results.
Exact values of special angles
| θ (deg) | θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 15° | π/12 | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 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 |
| 75° | 5π/12 | (√6 + √2)/4 | (√6 − √2)/4 | 2 + √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
Key identities
| Identity | Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ |
| Reciprocal | csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ |
| Quotient | tan θ = sin θ/cos θ, cot θ = cos θ/sin θ |
| Co-function | sin(90° − θ) = cos θ, tan(90° − θ) = cot θ |
| Negative angle | sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ |
| Double angle | sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ |
Frequently asked questions
What are the six trigonometric functions?
For an angle θ with point (x, y) on the unit circle: sin θ = y, cos θ = x, tan θ = y/x, and their reciprocals csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. In a right triangle, sin = opposite/hypotenuse, cos = adjacent/hypotenuse and tan = opposite/adjacent.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, and radians by 180/π to get degrees. So 30° = π/6 ≈ 0.5236 rad and 1 rad ≈ 57.2958°. A full circle is 360° = 2π rad.
Why is tan 90° undefined?
tan θ = sin θ ÷ cos θ, and cos 90° = 0, so the division is undefined. The same happens at 270° and every 180° from there. Likewise csc and cot are undefined where sin θ = 0 (0°, 180°, 360°…), and sec where cos θ = 0.
What are the exact values of special angles?
sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, and cosines are the same in reverse order. tan 30° = √3/3, tan 45° = 1, tan 60° = √3. For 15° and 75°, sin 15° = (√6 − √2)/4 and tan 15° = 2 − √3.
What is the range of arcsin, arccos and arctan?
To be functions, inverses return one principal value: arcsin gives −90° to 90°, arccos gives 0° to 180°, and arctan gives values strictly between −90° and 90°. This calculator uses 0° to 180° for arccot. Other angles with the same sine or cosine are listed as additional solutions.
What is a reference angle?
The reference angle is the acute angle between the terminal side and the x-axis. Trig values of any angle equal ± the values of its reference angle; the sign depends on the quadrant (All Students Take Calculus: all positive in I, sin in II, tan in III, cos in IV).