Math July 13, 2026 · 8 Min Read

Arcsin Calculator – Guide & Formulas

Calculate arcsin (inverse sine) instantly. Enter a sine value to find the angle in degrees or radians with step-by-step results.

The Arcsin Calculator computes the inverse sine of any value between -1 and 1. Enter a sine value and instantly receive the angle in both degrees and radians, along with a step-by-step explanation of how the inverse sine function works.

Key Takeaway

Use the free Arcsin Calculator to calculate arcsin (inverse sine) instantly. enter a sine value to find the angle in degrees or radians with step-by-step results. Get instant results with step-by-step explanations.

How to Use the Arcsin Calculator

  1. Enter a sine value between -1 and 1 in the input field.
  2. The calculator instantly computes the arcsin result.
  3. View the angle in both degrees and radians.
  4. Review the step-by-step explanation and formula reference.

The Formula

arcsin(x) = θ where sin(θ) = x and -π/2 ≤ θ ≤ π/2 (or -90° ≤ θ ≤ 90°)

Variable Definitions

  • arcsin(x): The angle whose sine is x, also written as sin⁻¹(x)
  • x: The sine value (input), must be between -1 and 1
  • θ: The resulting angle in radians (-π/2 to π/2) or degrees (-90° to 90°)
  • sin(θ): The sine function: opposite/hypotenuse in a right triangle

Example: Arcsin(0.5)

Find the angle whose sine is 0.5.

  1. Step 1: We need to find θ such that sin(θ) = 0.5.
  2. Step 2: From the unit circle or sine table, sin(30°) = sin(π/6) = 0.5.
  3. Step 3: Therefore arcsin(0.5) = 30° or π/6 radians ≈ 0.5236 radians.
  4. Step 4: Verify: sin(30°) = 0.5 ✓

Frequently Asked Questions

What is arcsin?

Arcsin (inverse sine) is the function that takes a sine value and returns the angle whose sine is that value. It is written as arcsin(x) or sin⁻¹(x) and produces angles between -90° and 90° (-π/2 to π/2 radians).

What is the domain and range of arcsin?

The domain of arcsin is [-1, 1], meaning the input must be between -1 and 1 inclusive. The range is [-π/2, π/2] in radians or [-90°, 90°] in degrees.

How is arcsin different from 1/sin?

Arcsin (sin⁻¹) is the inverse function that returns an angle, while 1/sin is the cosecant function (csc) which returns a ratio. They are completely different operations.

What is arcsin(0)?

arcsin(0) = 0 radians = 0°. This is because sin(0°) = 0. The angle whose sine is 0 is zero radians.

What is arcsin(1)?

arcsin(1) = π/2 radians = 90°. This is because sin(90°) = 1. The angle whose sine is 1 is a right angle.

What is arcsin(-1)?

arcsin(-1) = -π/2 radians = -90°. This is because sin(-90°) = -1. The angle whose sine is -1 is -90°.

Can arcsin return angles outside [-90°, 90°]?

No. By definition, the principal value of arcsin always returns angles in the range [-π/2, π/2] radians or [-90°, 90°] degrees.

What are common arcsin values?

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

How do I convert arcsin result from radians to degrees?

Multiply the radian result by 180/π to convert to degrees. For example, arcsin(0.5) = π/6 radians × (180/π) = 30°.

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°.