Inverse Trig Calculator — Arcsin, Arccos
Calculate arcsin, arccos, arctan and get angles from trig ratios.
Compute inverse trigonometric functions (arcsin, arccos, arctan)
What Is the Inverse Trig Calculator — Arcsin, Arccos?
The Inverse Trigonometry Calculator computes arcsin, arccos, and arctan — the inverse functions of sine, cosine, and tangent. Given a trig ratio, the inverse function returns the angle that produces it. Results are given in both degrees and radians.
Formula
How to Use
Select the inverse function (arcsin, arccos, or arctan) from the dropdown. Enter the value and click Calculate. For arcsin and arccos, the input must be between −1 and 1. For arctan, any real number is valid.
Example Calculation
arcsin(0.5): sin(30°) = 0.5, so arcsin(0.5) = 30° = π/6 rad arccos(0): cos(90°) = 0, so arccos(0) = 90° = π/2 rad arctan(1): tan(45°) = 1, so arctan(1) = 45° = π/4 rad arctan(√3) = 60° = π/3 rad
Understanding Inverse Trig — Arcsin, Arccos
Inverse trigonometric functions are essential tools in mathematics, engineering, and physics. They answer the question: "What angle has this trigonometric ratio?"
In physics, arctan is used to find the direction of a resultant vector from its components. In optics, arcsin appears in Snell's Law for calculating refraction angles. In computer graphics, atan2 (a variant of arctan) is ubiquitous for computing rotation angles.
The notation arcsin(x) and sin⁻¹(x) are equivalent. Be careful: sin⁻¹(x) does NOT mean 1/sin(x) — that is the cosecant function csc(x). The "−1" superscript denotes the inverse function, not the reciprocal.
Frequently Asked Questions
Why is the domain of arcsin limited to [−1, 1]?
The sine function only produces values between −1 and 1. Since arcsin is the inverse of sin, it can only accept inputs in that range.
Why does arccos return values in [0°, 180°] only?
To be a function, arccos must give exactly one output per input. By convention (the "principal value"), it is restricted to [0°, 180°], where cosine is one-to-one.
What is the difference between arctan and atan2?
arctan(y/x) has an ambiguity in the quadrant because dividing y by x loses the sign of each. atan2(y, x) takes both coordinates and returns the angle in the correct quadrant, from −180° to 180°.
What are some real-world uses of inverse trig?
Finding angles in engineering (the angle of a ramp, the angle of elevation to a building), physics (angle of incidence of light), and navigation (bearing calculations).