Quick answer
Choose arcsin, arccos or arctan and enter a value. arcsin(0.5) = 30° (0.5236 rad). arcsin and arccos require −1 ≤ x ≤ 1; arctan accepts any value.
Pick the inverse function and enter x. The calculator returns the angle in degrees and radians. Use this inverse trigonometry calculator to check your own figures.
Related tools: Trigonometry Calculator: All six trigonometric functions of an angle. Inverse Sine Calculator: arcsin (inverse trigonometric function) with the result in degrees and radians. Inverse Cosine Calculator: arccos (inverse trigonometric function) with the result in degrees and radians.
How to use the Inverse Trigonometry Calculator
- Choose function: arcsin (sin⁻¹), arccos (cos⁻¹) and arctan (tan⁻¹).
- Enter value.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Function | Choose one: arcsin (sin⁻¹); arccos (cos⁻¹); arctan (tan⁻¹) | Starts at arcsin (sin⁻¹) |
| Value | A number | — |
Formula and method
Formula
θ = sin⁻¹(x), cos⁻¹(x) or tan⁻¹(x)
IEEE-754 double precision; shown to the decimals indicated.
Principal values: arcsin −90° to 90°, arccos 0° to 180°, arctan −90° to 90°.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Function = arcsin (sin⁻¹); Value = 0.5.
| Result | Value |
|---|---|
| Angle | 30° |
| Radians | 0.523599 |
Example 2
Inputs: Function = arccos (cos⁻¹); Value = -0.5.
| Result | Value |
|---|---|
| Angle | 120° |
| Radians | 2.094395 |
Example 3
Inputs: Function = arctan (tan⁻¹); Value = 1.
| Result | Value |
|---|---|
| Angle | 45° |
| Radians | 0.785398 |
The first line is the headline result (angle). Below it the inverse Trigonometry Calculator lists radians, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
The three functions
arcsin returns −90° to 90°, arccos 0° to 180° and arctan −90° to 90° (exclusive). They undo sin, cos and tan within these principal ranges.
Identity
arcsin(x) + arccos(x) = 90° for any x in range: 30° + 60° = 90° for x = 0.5. Use the single-function pages for arccos, arcsin and arctan if you only need one.
How the result changes with value
Holding every other input at the first worked example, this table changes only value. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Value | Angle | Radians |
|---|---|---|
| 0.25 | 14.477512° | 0.25268 |
| 0.5 | 30° | 0.523599 |
| 1 | 90° | 1.570796 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Value: entering "2" shows "The value must be between −1 and 1." beside that field and no result.
Choosing between the Inverse Trigonometry Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Trigonometry Calculator](/math/trigonometry-calculator/) | angle and unit | sin |
| [Inverse Sine Calculator](/math/inverse-sine/) | value | arcsin(0.5) |
| [Inverse Cosine Calculator](/math/inverse-cosine-calculator/) | value | arccos(0.5) |
| [Inverse Tangent Calculator](/math/inverse-tangent-calculator/) | value | arctan(1) |
| [SOHCAHTOA Calculator](/math/sohcahtoa-calculator/) | acute angle, known side length and known side is the | Opposite |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- arcsin and arccos need −1 ≤ x ≤ 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for value to confirm this.
Frequently asked questions
What does sin⁻¹ mean?
The inverse sine, not 1/sin.
What is cos⁻¹(0)?
cos⁻¹(0) = 90°, the angle whose cosine is 0.
Can I get radians?
Yes, both are shown.
What does the Inverse Trigonometry Calculator calculate?
Arcsine, arccosine or arctangent in degrees and radians. It returns Angle and Radians.
What formula does the Inverse Trigonometry Calculator use?
θ = sin⁻¹(x), cos⁻¹(x) or tan⁻¹(x). Principal values: arcsin −90° to 90°, arccos 0° to 180°, arctan −90° to 90°.
What do I need to enter in the inverse Trigonometry Calculator?
You enter Function and Value.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. arcsin and arccos need −1 ≤ x ≤ 1.
What does the “Function” option change?
It sets how the inverse Trigonometry is worked out. The choices are arcsin (sin⁻¹), arccos (cos⁻¹) and arctan (tan⁻¹).
What is an example of a inverse Trigonometry result?
With function arcsin (sin⁻¹) and value 0.5, the angle is 30°.
Where does the method come from?
The method follows: OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions; OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry.
Related calculators
- Trigonometry Calculator — All six trigonometric functions of an angle.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
- Inverse Cosine Calculator — arccos (inverse trigonometric function) with the result in degrees and radians.
- Inverse Tangent Calculator — arctan (inverse trigonometric function) with the result in degrees and radians.
- SOHCAHTOA Calculator — Find the other two sides of a right triangle from one side and an acute angle.
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.