Inverse Trigonometric Calculator

Background.

A wheelchair ramp built to the 1:12 maximum slope that US accessibility standards allow rises one unit for every twelve it runs. What angle is that? The ratio is the tangent of the slope angle, so the question runs trigonometry backwards: given a ratio, find the angle. Arctangent answers it, at about 4.76°. Arcsine and arccosine do the same job for the other two ratios, opposite over hypotenuse and adjacent over hypotenuse.

Running a function backwards creates a problem the forward direction never has. The sine of 30° is 0.5, but so is the sine of 150°, and of 390°, and of every angle that differs from those by a full turn. An inverse function has to return one number, so a principal range was agreed for each: arcsine gives angles from −90° to 90°, arccosine from 0° to 180°, and arctangent from just above −90° to just below 90°. Inside those ranges every ratio matches exactly one angle.

The ranges suit the jobs the functions usually do. Arccosine's 0° to 180° is exactly the span of possible angles between two directions, which is why the angle between two vectors comes from the arccosine of their dot product divided by both lengths. Arcsine and arctangent are centred on zero, so a negative ratio gives a negative angle: a downhill grade, a dip below the horizontal.

What a principal value cannot tell you is which quadrant a direction really lies in. Arctangent of y ÷ x sees only the ratio, so the points (1, 1) and (−1, −1) both come out at 45°, although the second points the opposite way. Programming languages provide a two-argument atan2 for that reason. This page gives the principal angle only, in degrees and radians, and leaves any other solutions to you.

What is inverse trigonometric calculator?

An inverse trigonometric function takes a ratio and returns an angle. Arcsine, written sin⁻¹, gives the angle whose sine is x; arccosine, cos⁻¹, the angle whose cosine is x; and arctangent, tan⁻¹, the angle whose tangent is x. Because sine, cosine and tangent repeat endlessly, each inverse is restricted to a principal range so that it returns a single answer. Arcsine and arccosine accept x only from −1 to 1, while arctangent accepts any real number.

How to use this calculator.

1. 01Choose the function that matches the ratio you have: arcsin for opposite ÷ hypotenuse, arccos for adjacent ÷ hypotenuse, arctan for opposite ÷ adjacent, which is rise over run.
2. 02Enter the ratio itself as a decimal, not an angle. A roof that rises 6 inches in 12 is a ratio of 0.5, entered with arctan.
3. 03For arcsin and arccos the value must lie between −1 and 1. A larger number means a side was measured wrongly or the division was done the wrong way round.
4. 04Read the angle in degrees, or take the radian figure if it is going into a formula or spreadsheet that works in radians.
5. 05If your problem allows angles outside the principal range, build them from the result: 180° minus it for sine, its negative for cosine, plus or minus 180° for tangent, adding whole turns of 360° as needed.

The formula.

θ = arcsin(x), arccos(x) or arctan(x). The angle is first produced in radians and then converted to degrees by multiplying by 180 and dividing by π. The principal ranges are −π/2 to π/2 for arcsin, 0 to π for arccos, and the open interval between −π/2 and π/2 for arctan; in degrees, −90° to 90°, 0° to 180°, and strictly between −90° and 90°. Arctangent never reaches ±90° because the tangent is undefined there; very large inputs only creep towards it, so x = 1,000,000 gives about 89.99994°. Near the ends of their domain, arcsine and arccosine become very sensitive: arcsin(0.99) is about 81.89° while arcsin(0.999) is about 87.44°, so a ratio measured slightly high near 1 can move the angle by several degrees. Both outputs are rounded to ten decimal places.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.