Inverse Trig Calculator (arcsin, arccos, arctan)

Going backwards from a ratio to an angle

Regular trigonometric functions take an angle and give you a ratio: sin(30°) = 0.5. The inverse functions run that in reverse. You hand them a ratio and they hand back the angle, so arcsin(0.5) = 30°. They are written arcsin, arccos, arctan, or sometimes sin⁻¹, cos⁻¹, tan⁻¹. That little −1 is notation for “inverse,” not a reciprocal, and mixing those two up trips a lot of students.

This calculator takes a value, applies the inverse function you pick, and returns the angle in both degrees and radians.

The three functions and their ranges

Because sine, cosine, and tangent repeat forever, infinitely many angles share the same ratio. To return a single answer, each inverse function is restricted to a principal range:

• arcsin(x): input from −1 to 1, output from −90° to 90°
• arccos(x): input from −1 to 1, output from 0° to 180°
• arctan(x): input any real number, output from −90° to 90°

arcsin and arccos reject anything outside [−1, 1], because no real angle has a sine or cosine larger than 1. arctan accepts every value, since the tangent runs off to infinity.

A worked example

A ramp rises 3 m over a horizontal run of 8 m. Its angle of incline is arctan(3/8) = arctan(0.375), about 20.6°. That is far steeper than the 4.8° maximum most accessibility codes allow for a wheelchair ramp, so this slope would fail inspection.

Identities worth knowing

arcsin(x) + arccos(x) = 90° for every x in [−1, 1], so once you have one, the other is just 90° minus it. For positive x, arctan(x) + arctan(1/x) = 90° as well.

The arctan quadrant trap

arctan only returns angles between −90° and 90°, so it cannot tell a first-quadrant point from a third-quadrant one: both (1, 1) and (−1, −1) give 45°. Programmers use the two-argument atan2(y, x) instead, which reads the signs of both coordinates and returns the correct angle in all four quadrants. If you are turning coordinates into an angle in code, reach for atan2, not plain arctan.

Where they show up

Finding angles from side lengths in a right triangle, recovering a launch angle in physics, inverse kinematics in robotics, pulling Euler angles out of a rotation matrix in 3D graphics, and converting a coordinate difference into a compass bearing.