Inverse Trigonometric Functions Calculator
Find an unknown angle from a known trigonometric ratio using arcsine, arccosine, or arctangent. Useful in right-triangle problems, physics, and engineering when the angle itself is the unknown.
About this calculator
Inverse trigonometric functions reverse the standard sin, cos, and tan operations to recover an angle from a ratio. Arcsine (arcsin) answers: 'which angle has this sine value?' — written θ = arcsin(x). Similarly, arccos(x) finds the angle whose cosine equals x, and arctan(x) finds the angle whose tangent equals x. The domain for arcsin and arccos is [−1, 1], while arctan accepts any real number. Results from arcsin lie in [−90°, 90°], from arccos in [0°, 180°], and from arctan in (−90°, 90°). The calculator applies JavaScript's built-in Math.asin, Math.acos, or Math.atan functions (which return radians) and optionally multiplies by 180/π to convert to degrees.
How to use
Suppose you want to find the angle whose sine is 0.5 (arcsin(0.5)) in degrees. Enter 0.5 as the trigonometric ratio, select 'arcsin' as the function, and choose 'degrees' as the output unit. The calculator computes: θ = arcsin(0.5) × 180/π = 0.5236 × 180/π ≈ 30°. So the angle is 30°. For arctan(1), the result would be arctan(1) × 180/π ≈ 45°. If you select radians, the output is returned directly from Math.asin/acos/atan without conversion.
Frequently asked questions
What is the difference between arcsin, arccos, and arctan?
Arcsin finds the angle whose sine equals a given value; arccos finds the angle whose cosine equals a given value; arctan finds the angle whose tangent equals a given value. They are the inverse operations of sin, cos, and tan respectively. Arcsin and arccos require inputs between −1 and 1 because sine and cosine can never exceed those bounds, while arctan accepts any real number. Each function returns a unique principal value within a specific range to ensure a single, unambiguous answer.
Why is the domain of arcsin and arccos restricted to values between -1 and 1?
Sine and cosine functions only ever produce values in the range [−1, 1], so it is impossible for any angle to have a sine or cosine outside that interval. Inputting a value outside [−1, 1] into arcsin or arccos has no real solution and results in NaN (not a number). This is a fundamental property of the unit circle, where the x-coordinate (cosine) and y-coordinate (sine) cannot exceed 1 in absolute value. Arctangent has no such restriction because the tangent function spans all real numbers.
How do I use inverse trigonometric functions to find a missing angle in a right triangle?
If you know any two sides of a right triangle, you can form a ratio and apply the appropriate inverse function. For example, if the opposite side is 3 and the hypotenuse is 5, the sine ratio is 3/5 = 0.6, so the angle = arcsin(0.6) ≈ 36.87°. If you know the adjacent and hypotenuse, use arccos; if you know opposite and adjacent, use arctan. Enter the computed ratio into this calculator, select the correct function and output unit, and the angle is returned immediately.