📐 Trigonometric Calculator (Radian)
Calculate trigonometric function values based on radians in an equilateral triangle.
Radian-Based Trigonometric Functions
You can enter the angle directly in radians (rad) or in the π/n form.
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent = sin θ / cos θ
Radian and Degree Conversion
1 rad = 180/π ≈ 57.296°
1° = π/180 ≈ 0.017453 rad
Key Radian Values
| rad | π form | Deg | sin |
|---|---|---|---|
| 0 | 0 | 0° | 0 |
| 0.524 | π/6 | 30° | 0.5 |
| 0.785 | π/4 | 45° | √2/2 |
| 1.047 | π/3 | 60° | √3/2 |
| 1.571 | π/2 | 90° | 1 |
| 3.142 | π | 180° | 0 |
Euler's Formula
e^(iθ) = cos θ + i·sin θ
Using radians, Euler's formula holds naturally. When θ=π, e^(iπ) + 1 = 0.
Worked examples
At π/4 the sine and cosine are both 0.7071068 and the tangent is 1, exactly as at 45° in degrees.
At π/6 the sine is 0.5, the cosine 0.8660254 and the tangent 0.5773503, the same values as 30°.
What the values assume
Radians measure arc length on a unit circle, so a full turn is 2π ≈ 6.2831853 instead of 360. Tangent is undefined at π/2 and its odd multiples, which the calculator marks. In the π/n mode n divides π, so n = 4 means π/4 ≈ 0.7853982.
Frequently asked questions
- Why do mathematicians prefer radians?
- Only in radians is the derivative of sine equal to cosine with no extra constant, which is what makes the small-angle expansion sin x ≈ x − x³/6 valid.
- How do I convert degrees and radians?
- Multiply degrees by π/180, or radians by 180/π. So π/3 is 60° and one radian is about 57.2958°.
- What are the exact values at common angles?
- sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2 and sin(π/2) = 1, which the decimal output approximates.