Math July 13, 2026 · 8 Min Read

Inverse Trigonometric Functions Calculator – Guide & Formulas

Calculate all inverse trigonometric functions instantly. Find arcsin, arccos, arctan, arccsc, arcsec, and arccot values in degrees or radians.

The Inverse Trigonometric Functions Calculator computes all six inverse trigonometric functions: arcsin, arccos, arctan, arccsc, arcsec, and arccot. Enter a value to find the corresponding angle in both degrees and radians, with domain and range information for each function.

Key Takeaway

Use the free Inverse Trigonometric Functions Calculator to calculate all inverse trigonometric functions instantly. find arcsin, arccos, arctan, arccsc, arcsec, and arccot values in degrees or radians. Get instant results with step-by-step explanations.

How to Use the Inverse Trigonometric Functions Calculator

  1. Select the inverse trigonometric function you want to compute.
  2. Enter the input value within the valid domain for that function.
  3. Select degrees or radians as the output unit.
  4. View the angle result in both degrees and radians.
  5. Review the domain, range, and formula reference.

The Formula

arcsin(x) = θ where sin(θ) = x | arccos(x) = θ where cos(θ) = x | arctan(x) = θ where tan(θ) = x

Variable Definitions

  • arcsin(x): The angle whose sine is x, range [-π/2, π/2]
  • arccos(x): The angle whose cosine is x, range [0, π]
  • arctan(x): The angle whose tangent is x, range (-π/2, π/2)
  • arccsc(x): The angle whose cosecant is x, range [-π/2, π/2] \ {0}
  • arcsec(x): The angle whose secant is x, range [0, π] \ {π/2}
  • arccot(x): The angle whose cotangent is x, range (0, π)

Example: Compute all inverse trig values for x = 0.5

Find arcsin(0.5), arccos(0.5), and arctan(0.5).

  1. Step 1: arcsin(0.5): Find θ where sin(θ) = 0.5 → θ = 30° (π/6 rad).
  2. Step 2: arccos(0.5): Find θ where cos(θ) = 0.5 → θ = 60° (π/3 rad).
  3. Step 3: arctan(0.5): Find θ where tan(θ) = 0.5 → θ ≈ 26.565° (≈ 0.4636 rad).
  4. Step 4: Note: arcsin(x) + arccos(x) = 90° for any valid x.

Frequently Asked Questions

What are inverse trigonometric functions?

Inverse trigonometric functions are the inverses of the basic trigonometric functions. They take a ratio (like 0.5) as input and return an angle. For example, arcsin(0.5) = 30° because sin(30°) = 0.5. There are six inverse trig functions corresponding to the six trig functions.

What are the domains and ranges of inverse trig functions?

arcsin and arccsc: domain [-1, 1] and [-∞,-1]∪[1,∞], range [-π/2, π/2]. arccos and arcsec: domain [-1, 1] and [-∞,-1]∪[1,∞], range [0, π]. arctan: domain (-∞,∞), range (-π/2, π/2). arccot: domain (-∞,∞), range (0, π).

Why are there restrictions on the ranges?

Without range restrictions, inverse trig functions would not be proper functions (they would produce multiple outputs for one input). The restricted ranges are chosen to include the most commonly used angles while maintaining the function property.

What is the difference between arcsin and csc⁻¹?

arcsin (sin⁻¹) takes a sine value and returns the angle whose sine is that value. csc⁻¹ (arccsc) takes a cosecant value and returns the angle whose cosecant is that value. Since csc(θ) = 1/sin(θ), csc⁻¹(x) = sin⁻¹(1/x).

How do I compute arccsc(2)?

arccsc(2) = arcsin(1/2) = arcsin(0.5) = 30° (π/6 radians). Since csc(θ) = 1/sin(θ), you can convert any arccsc problem to an arcsin problem.

What is the relationship between arcsin and arccos?

The identity arcsin(x) + arccos(x) = π/2 (or 90°) relates the two functions. This means the angle whose sine is x plus the angle whose cosine is x always equals 90°. This follows from the cofunction identity.

Can inverse trig functions handle negative inputs?

Yes, for arcsin, arctan, and arccot. arcsin(-x) = -arcsin(x), arctan(-x) = -arctan(x). For arccos, the result is in [π/2, π] for negative inputs: arccos(-x) = π - arccos(x).

What are the most common inverse trig values?

Common values: arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(√2/2) = 45°, arcsin(√3/2) = 60°, arcsin(1) = 90°. arccos(1) = 0°, arccos(√3/2) = 30°, arccos(√2/2) = 45°, arccos(0.5) = 60°, arccos(0) = 90°.

How do I convert between degrees and radians?

Multiply radians by 180/π to get degrees. Multiply degrees by π/180 to get radians. For example, π/4 radians × (180/π) = 45°, and 30° × (π/180) = π/6 radians.

When would I use arccsc or arcsec?

arccsc and arcsec are less commonly used but appear in calculus integration formulas (e.g., ∫sec(x)dx = ln|sec(x) + tan(x)| + C) and in advanced trigonometric identities. They are also useful when working with secant and cosecant ratios directly.