Trigonometry Calculator
Sine, cosine, tangent, cosecant, secant and cotangent of any angle in degrees or radians, exact on the unit circle, or the angle back from a value.
Result
sin: 1/2 = 0.5
Exact on the unit-circle table; other values to ten significant figures.
- cos
- √3/2 ≈ 0.8660254038
- tan
- √3/3 ≈ 0.5773502692
- csc
- 2
- sec
- 2√3/3 ≈ 1.154700538
- cot
- √3 ≈ 1.732050808
- The angle
- 30° = π/6 rad ≈ 0.5235987756 rad
- Where it points
- Quadrant I
- Note
- An angle on the unit-circle table, so the values are exact.
https://calcgallery.com/geometry/trigonometry-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Trigonometry Calculator
- To get the six functions, type an angle and choose degrees or radians. In radians you can type pi/6 or 2π/3.
- To go back from a value to an angle, choose the inverse, pick sin⁻¹, cos⁻¹ or tan⁻¹, and enter the value.
- Read exact forms such as √3/2 where the angle is a multiple of 30° or 45°, and ten-figure decimals elsewhere.
The unit circle
Every value here comes from the point where a radius of length 1 ends after turning through the angle:
(x, y) = (cos t, sin t)tan t = sin t ÷ cos t sec t = 1 ÷ cos t (cos t ≠ 0)cot t = cos t ÷ sin t csc t = 1 ÷ sin t (sin t ≠ 0)Exact values
At 0°, 30°, 45°, 60° and 90° the point has exact coordinates, such as (√3/2, 1/2) at 30°. Every multiple of 30° or 45° has one of these as its reference angle, the acute angle to the x-axis, and differs only in sign: sine is negative below the x-axis, cosine is negative left of the y-axis. The page uses that table for those angles, so the answer is written exactly and never carries rounding noise.
Any other angle is computed numerically and shown to ten significant figures. An angle typed in degrees or as a multiple of π is first brought within one turn exactly, so 3,600,030° is as accurate as 30°.
Inverses
sin⁻¹ and cos⁻¹ only accept values from −1 to 1, since the point on the unit circle never goes further out than that; tan⁻¹ accepts any number. The inverse secant, cosecant and cotangent are left out because textbooks define their ranges in different ways. For the sides and angles of a right triangle, use the right triangle calculator, and for converting an angle between units, the degrees and radians converter.
With your numbers
- (cos t, sin t) is the point on the unit circle at angle t
- tan = sin ÷ cos, csc = 1 ÷ sin, sec = 1 ÷ cos, cot = cos ÷ sin
Quick reference
Exact on the unit-circle table; other values to ten significant figures.
| Example | Result |
|---|---|
| 30° | sin: 1/2 = 0.5 |
| 45° | sin: √2/2 ≈ 0.7071067812 |
| 120° | sin: √3/2 ≈ 0.8660254038 |
| 90° | sin: 1 |
| 37° | sin: 0.6018150232 |
| π/6 rad | sin: 1/2 = 0.5 |
| 1 rad | sin: 0.8414709848 |
| sin⁻¹ 0.5 | Angle, in degrees: 30° |
| cos⁻¹ −0.5 | Angle, in degrees: 120° |
| tan⁻¹ 2 | Angle, in degrees: 63.43494882° |
Common questions about the Trigonometry Calculator
- What are sin, cos and tan?
- Draw a circle of radius 1 centred at the origin and turn a radius through the angle from the positive x-axis. The point it reaches is (cos, sin). Tangent is sin ÷ cos, the slope of that radius. In a right triangle this matches the familiar ratios: opposite over hypotenuse, adjacent over hypotenuse, opposite over adjacent.
- What are csc, sec and cot?
- The reciprocals: cosecant is 1 ÷ sin, secant is 1 ÷ cos and cotangent is cos ÷ sin, which is 1 ÷ tan wherever tan is defined. At 30°, sin is 1/2, so csc is 2.
- Why does tan 90° say not defined?
- Because cos 90° is 0, and tan is sin ÷ cos. A floating-point calculator gets cos 90° as 6.1 × 10⁻¹⁷ instead of 0 and prints a tangent of about 1.6 × 10¹⁶. This page recognises 90° as a unit-circle angle, uses the exact values and says tan is not defined there, as are sec 90°, csc 0° and cot 0°.
- Why does sin⁻¹ 0.5 give only 30°, when 150° has the same sine?
- An inverse function has to give one answer, so each is limited to a range: sin⁻¹ gives angles from −90° to 90°, cos⁻¹ from 0° to 180° and tan⁻¹ between −90° and 90°. The one in range is the principal value. Other angles with the same sine are 180° minus it, and either of those plus any whole number of turns.
- Degrees or radians?
- Both measure the same turn: 360° is 2π radians, so 30° is π/6. Radians are the natural unit in calculus and in most programming languages, which is a common reason for a sine that looks wrong: sin 30 in radians is −0.988, not 0.5.