About the Inverse Trig Calculator
Sine, cosine and tangent turn an angle into a ratio. Their inverses, arcsin, arccos and arctan, turn a ratio back into an angle. If a ramp rises 1 metre for every 2 metres of its slanted length, arcsin of 0.5 tells you the ramp's angle is 30 degrees. This calculator takes the ratio, applies the inverse function you choose and gives the angle in degrees, radians, gradians and as a multiple of pi. It also lists a second angle in the full turn that has the same value, because many angles share a sine or cosine. Out-of-range values, such as an arcsin of 2, give a clear message.
Updated October 2026
How to use the Inverse Trig Calculator
- 1Choose arcsin, arccos or arctan.
- 2Enter the ratio. For arcsin and arccos it must be between −1 and 1.
- 3Read the principal angle in degrees and radians.
- 4Look at the other forms and the second angle with the same value.
- 5Copy the result.
Common values
- arcsin(0.5) = 30°, arccos(0.5) = 60°.
- arcsin(1) = 90°, arccos(0) = 90°.
- arctan(1) = 45°, arctan(0) = 0°.
- arcsin(√2/2) ≈ 45°, arccos(√3/2) = 30°.
- arctan(√3) = 60°.
Applications
Builders use inverse trig to find the pitch of a roof from its rise and run, surveyors to work out angles from measured distances, pilots and sailors for navigation bearings, and game developers to aim and rotate. Whenever you know two sides of a right triangle and need an angle, an inverse function does the job.
When you apply the result to a real problem, think about which angle you want. If a ladder leans against a wall and you know it reaches 3.5 metres up a 4 metre ladder, arcsin of 0.875 gives about 61 degrees from the ground, which is the one you want. The other solution, 119 degrees, would describe a ladder leaning the other way, so it makes no physical sense. Always sketch the situation and pick the answer that fits.
Cautions
Check that your calculator is working in degrees if you expect degrees. Rounding in the ratio affects angles near 90 degrees for arcsin and near 0 or 180 for arccos, since small changes in the ratio make big changes in the angle there. For arctan of a ratio of two sides where the run might be zero or negative, use an atan2 style calculation to get the correct quadrant.
Related tools
The Trigonometry Calculator finds sine, cosine and tangent from angles, the Triangle Calculator solves whole triangles, the Angle Converter changes between units and the Pythagorean Theorem Calculator finds missing sides.
Features
- Arcsin, arccos and arctan
- Degrees, radians, gradians and multiples of pi
- Second solution in a full turn
- Domain checks with plain messages
- High precision
- Works offline in your browser
Frequently asked questions
What is arcsin?
Arcsin, also written sin⁻¹, gives the angle whose sine is a given number. Arcsin of 0.5 is 30 degrees, because the sine of 30 degrees is 0.5. It accepts inputs from −1 to 1.
Is sin⁻¹ the same as 1/sin?
No. The superscript −1 means inverse function, not reciprocal. The reciprocal of sine is cosecant. To avoid confusion, many textbooks write arcsin instead.
Why are there two angles with the same sine?
Sine is positive in both the first and second quadrants. Between 0 and 360 degrees, an angle of 30 degrees and one of 150 degrees both have a sine of 0.5. The calculator gives the principal value and the second one.
What range do the inverse functions return?
Arcsin and arctan return angles from −90 to 90 degrees. Arccos returns angles from 0 to 180 degrees. These are the principal values used by calculators and programming languages.
How do I use arctan to find the slope angle?
Divide the rise by the run to get the slope, then take the arctan. A slope of 1 (a 45 degree angle) has arctan 45 degrees. A 1 in 12 ramp has an angle of about 4.76 degrees.
How do I convert radians to degrees?
Multiply by 180 and divide by pi. The calculator already shows both, and also the angle as a multiple of pi, such as 0.1667π for 30 degrees.