Inverse Trig Calculator — Arcsin, Arccos, Arctan Values

The Formula

Inverse trigonometric functions reverse the action of the standard trigonometric functions. If y = sin(θ), then θ = arcsin(y) returns the angle whose sine is y. Because trigonometric functions are periodic and not one-to-one over their entire domain, the inverse functions are defined on restricted domains to produce a unique principal value. arcsin has domain [-1, 1] and range [-90°, 90°]; arccos has domain [-1, 1] and range [0°, 180°]; arctan has domain all real numbers and range [-90°, 90°].

Variable Definitions

How to Use This Calculator

1. 1

   Select the inverse trigonometric function you want to compute: arcsin (inverse sine), arccos (inverse cosine), or arctan (inverse tangent).

2. 2

   Enter a numeric value. For arcsin and arccos, the value must be in the domain [-1, 1]. For arctan, any real number is accepted.

3. 3

   The calculator returns the principal value in both degrees and radians. The principal value is the unique angle within the function's restricted range.

4. 4

   If the input is outside the valid domain, a domain error message is shown explaining the restriction.

Quick Reference

Common Applications

• Solving trigonometric equations: finding unknown angles from known trigonometric ratios in geometry and physics problems
• Calculus and integration: inverse trig functions appear as antiderivatives of rational functions involving square roots, such as ∫(1/√(1-x²))dx = arcsin(x) + C
• Computer graphics: converting between coordinate representations, such as finding the angle of a vector from its x and y components using arctan2 (related to arctan)
• Navigation and robotics: computing heading angles from sensor readings and determining joint angles in robotic arm kinematics using inverse kinematics equations
• Signal processing: phase angle calculation in Fourier analysis, where the phase of a complex frequency component is found using arctan of the imaginary part over the real part

Principal values of inverse trigonometric functions. arcsin and arctan range from -90° to 90°, while arccos ranges from 0° to 180°. arctan has horizontal asymptotes at ±90°.

Understanding the Concept

Inverse trigonometric functions, also called arcus functions or cyclometric functions, are the inverse operations of the standard trigonometric functions sine, cosine, and tangent. They answer the question: given a trigonometric ratio, what angle produces it? For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30 degrees (π/6 radians). Because trigonometric functions are periodic — they repeat their values at regular intervals — they are not one-to-one over their full domains. A one-to-one function is required for a well-defined inverse. To resolve this, each inverse trig function is defined on a restricted domain of the original function. For sine, the restricted domain is [-90°, 90°] (or [-π/2, π/2] radians), where sine is increasing and takes all values from -1 to 1 exactly once. The inverse of this restricted sine is arcsin (also written as sin⁻¹). For cosine, the restricted domain is [0°, 180°] (or [0, π] radians), where cosine is decreasing and covers all values from 1 to -1. The inverse of this restricted cosine is arccos (cos⁻¹). For tangent, the restricted domain is (-90°, 90°) (or (-π/2, π/2) radians), where tangent is increasing and takes all real values. The inverse of this restricted tangent is arctan (tan⁻¹). These restricted ranges are called the principal values. For example, arcsin(0.5) = 30° is the principal value, even though sin(150°) = 0.5 and sin(390°) = 0.5 as well. The principal value is always the unique angle within the restricted range. The notation sin⁻¹(x) is common but potentially confusing because it resembles 1/sin(x) = csc(x). To avoid ambiguity, the arcsin notation is preferred, where "arc" means "the arc (angle) whose sine is." The inverse trig functions have important relationships. For any x in [-1, 1]: arcsin(x) + arccos(x) = 90° (or π/2 radians), because the sine of an angle equals the cosine of its complement. The derivatives of inverse trig functions are important in calculus: d/dx arcsin(x) = 1/√(1-x²), d/dx arccos(x) = -1/√(1-x²), and d/dx arctan(x) = 1/(1+x²). These derivatives appear frequently in integration problems, making inverse trig functions essential tools in calculus.

Frequently Asked Questions

Sources & References