Inverse Trigonometric Functions Calculator
Find the angle whose sine, cosine, or tangent equals a given value. Use it to recover angles from ratios in geometry, physics, and engineering problems.
About this calculator
Inverse trigonometric functions reverse the standard trig operations. Arcsine (arcsin or sin⁻¹) returns the angle θ such that sin(θ) = x, for x in [−1, 1], with output in [−90°, 90°]. Arccosine (arccos) returns θ such that cos(θ) = x, for x in [−1, 1], output in [0°, 180°]. Arctangent (arctan) returns θ such that tan(θ) = x, for any real x, output in (−90°, 90°). These are the principal values. A two-argument arctangent (atan2) uses both a y and x component to determine the full-quadrant angle, resolving the ambiguity of standard arctan. Output can be given in degrees or radians. These functions are essential for calculating angles in right triangles, vector directions, and inverse kinematics.
How to use
Suppose sin(θ) = 0.6 and you need to find θ. Select function type 'arcsin' and enter input value = 0.6. In degrees mode, the calculator computes θ = arcsin(0.6) = 180/π × asin(0.6) ≈ 36.87°. Verify: sin(36.87°) ≈ 0.600 ✓. For arctan with input = 1.0, the result is arctan(1) = 45° (since tan(45°) = 1). For arccos(0.5), the result is 60° (since cos(60°) = 0.5). Select radians to get results in π units instead.
Frequently asked questions
What is the domain and range of arcsin, arccos, and arctan functions?
Arcsin accepts inputs in [−1, 1] and returns angles in [−90°, 90°] (principal values). Arccos also accepts [−1, 1] but returns angles in [0°, 180°]. Arctangent accepts any real number and returns angles in (−90°, 90°). These restricted ranges are necessary because sine, cosine, and tangent are not one-to-one over their full domains, so the inverse functions must be defined over a limited output range to be mathematically valid. Inputs outside the valid domain will produce undefined or complex results.
Why does arctangent not always give the correct quadrant for an angle?
Standard arctangent (atan) only returns angles between −90° and 90°, so it cannot distinguish between angles in the second and fourth quadrants or the first and third. For example, atan(1) = 45°, but tan(225°) is also 1. The two-argument function atan2(y, x) resolves this by considering the signs of both the opposite and adjacent sides separately, returning the full four-quadrant angle in (−180°, 180°]. This is widely used in programming, robotics, and navigation to determine the correct bearing or heading.
How do I use inverse trigonometric functions to find angles in a right triangle?
Given a right triangle, if you know any two sides you can find either non-right angle. If the opposite side is 3 and the hypotenuse is 5, the angle is arcsin(3/5) = arcsin(0.6) ≈ 36.87°. If the adjacent side is 4 and hypotenuse is 5, the angle is arccos(4/5) = arccos(0.8) ≈ 36.87°. If the opposite side is 3 and adjacent is 4, use arctan(3/4) = arctan(0.75) ≈ 36.87°. All three give the same angle, confirming the 3-4-5 Pythagorean triple.