This online calculator allows you to find an angle from a known value of a trigonometric function — calculating the arcsine (arcsin), arccosine (arccos), arctangent (arctan), arccotangent (arccot), as well as arcsecant (arcsec) and arccosecant (arccsc). Enter the function value in the form below, choose whether you want to receive the result in degrees or radians, and the calculator will display the angle immediately.
Inverse trigonometric functions are frequently used in geometry, physics, and engineering calculations when the side ratios of a triangle are known and the angle itself needs to be determined. If, conversely, you need to calculate the function value from a known angle, please use our trigonometric calculator.
Calculator
How to Use the Calculator
- Enter the value of the trigonometric function in the input field.
- Select which inverse function to calculate — arcsin, arccos, arctan, arccot, as well as arcsec or arccsc.
- Select the result measurement unit — degrees or radians.
- Click the "Calculate" button — the calculator will display the angle value.
What are Inverse Trigonometric Functions
Inverse trigonometric functions (arc functions) perform the opposite action of standard trigonometric functions: if sin(α) = x, then arcsin(x) = α. They allow you to find the angle when the sine, cosine, tangent, or cotangent value of that angle is known.
- arcsin(x) — the angle whose sine is x
- arccos(x) — the angle whose cosine is x
- arctan(x) — the angle whose tangent is x
- arccot(x) — the angle whose cotangent is x
Domain and Range of Arc Functions
Since the sine and cosine of an angle are always within the range of -1 to 1, their inverse functions — arcsin and arccos — are defined only for arguments from -1 to 1. Attempting to calculate, for example, arcsin(2), has no solution in real numbers.
| Function | Domain (x) | Range (angle) |
|---|---|---|
| arcsin(x) | from -1 to 1 | from -90° to 90° (from -π/2 to π/2) |
| arccos(x) | from -1 to 1 | from 0° to 180° (from 0 to π) |
| arctan(x) | any real number | from -90° to 90° (from -π/2 to π/2), excluding boundaries |
| arccot(x) | any real number | from 0° to 180° (from 0 to π), excluding boundaries |
This explains why the result of each arc function always falls within the same range of angles (known as the principal value range), even if the angle in a real-world problem lies outside this range — in which case additional reduction formulas must be applied to the result.
Useful Formulas and Properties
- arcsin(x) + arccos(x) = 90° (π/2) — for any x in the interval [-1; 1]
- arctan(x) + arccot(x) = 90° (π/2) — for any real x
- arcsin(-x) = -arcsin(x) — odd function
- arccos(-x) = 180° - arccos(x)
- arctan(-x) = -arctan(x) — odd function
Table of Inverse Trigonometric Calculator Values for Main Arguments
| x | arcsin(x) | arccos(x) | arctan(x) | arccot(x) |
|---|---|---|---|---|
| -1 | -90° | 180° | -45° | 135° |
| -√3/2 | -60° | 150° | -60° | 150° |
| -√2/2 | -45° | 135° | -35.26° | 125.26° |
| -1/2 | -30° | 120° | -26.57° | 116.57° |
| 0 | 0° | 90° | 0° | 90° |
| 1/2 | 30° | 60° | 26.57° | 63.43° |
| √2/2 | 45° | 45° | 35.26° | 54.74° |
| √3/2 | 60° | 30° | 60° | 30° |
| 1 | 90° | 0° | 45° | 45° |
Values for arctan and arccot are provided for x = ±1, ±√3/2, ±√2/2, ±1/2 and rounded to hundredths.
Calculation Examples
Example 1. Calculating arcsin
Given x = 0.5.
arcsin(0.5) = 30°
Example 2. Calculating arccos for a negative value
Given x = -0.5.
arccos(-0.5) = 120°
Note: the result of arccos for a negative argument is always greater than 90°, because the range of arccos is from 0° to 180°.
Example 3. Calculating arctan
Given x = 1.
arctan(1) = 45°
Example 4. Verifying the property arcsin(x) + arccos(x) = 90°
Given x = 0.5.
arcsin(0.5) + arccos(0.5) = 30° + 60° = 90° ✓
Frequently Asked Questions
Why can't arcsin(2) be calculated?
Because the sine of an angle is always in the range from -1 to 1, and therefore arcsine is only defined for arguments in this interval. The number 2 is outside these boundaries, so there is no solution in real numbers.
In what range does the result of arccos lie?
The result of arccos is always within the range of 0° to 180° (from 0 to π radians), regardless of the sign of the input value.
What is the difference between arctan and arccot?
Both functions are related by the equation arctan(x) + arccot(x) = 90°. The range of arctan is from -90° to 90°, and arccot is from 0° to 180°. Both are defined for any real number.
How do I find all angles, not just the principal value?
The inverse trigonometric function returns only one angle from the principal value range. If you need to find all angles satisfying the equation (e.g., sin(x) = 0.5 in the range from 0° to 360°), you must add the corresponding period to the principal value and account for the function's symmetry.
How do I calculate the value of a trigonometric function from a known angle?
For this, the standard trigonometric function is used — sin, cos, tan, or cot. You can calculate trigonometric functions from an angle on our site.