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Arcsin Calculator (Inverse Sine)

Calculate the inverse sine (arcsin) of a value. Returns the angle whose sine equals the input.

Reviewed by Chase FloiedUpdated

This free online arcsin calculator (inverse sine) provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

The sine value (must be between -1 and 1)

Results

Angle (degrees)

30°

Angle (radians)

0.523599 rad

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Arcsin Calculator (Inverse Sine). Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Arcsin Calculator (Inverse Sine) instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Arcsin Calculator (Inverse Sine) Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Arcsin Calculator (Inverse Sine) when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Arcsin Calculator (Inverse Sine) is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the inverse sine (arcsin) of a value. Returns the angle whose sine equals the input. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Arcsin Calculator (Inverse Sine)

The arcsine function (also written sin⁻¹ or arcsin) is the inverse of the sine function. Given a value x between −1 and 1, arcsin(x) returns the angle θ whose sine equals x. The result is always in the range [−90°, 90°] or [−π/2, π/2] radians, called the principal value. This function is essential when you need to find an angle from a known ratio. For example, if you know the height and hypotenuse of a right triangle and need the angle, you use arcsin(height/hypotenuse). Arcsine appears in physics (calculating angles of refraction using Snell's law), engineering (determining phase angles), and statistics (the arcsine transformation). This calculator provides both degree and radian results.

The Math Behind It

The sine function is not one-to-one over its entire domain, so to define an inverse, we restrict its domain to [−π/2, π/2], where it is strictly increasing and covers the full range [−1, 1]. The resulting inverse function arcsin: [−1, 1] → [−π/2, π/2] is continuous and differentiable on (−1, 1). Its derivative is d/dx arcsin(x) = 1/√(1 − x²), which is important in integral calculus: ∫ dx/√(1 − x²) = arcsin(x) + C. The Taylor series is arcsin(x) = x + x³/6 + 3x⁵/40 + 15x⁷/336 + ..., converging for |x| ≤ 1. Near x = 0, arcsin(x) ≈ x (the small angle approximation). The arcsine distribution is a probability distribution with PDF f(x) = 1/(π√(x(1−x))) on (0, 1), which arises in random walks and Brownian motion. In the complex plane, arcsin(z) = −i ln(iz + √(1 − z²)), extending the function to all complex numbers.

Formula Reference

Inverse Sine

θ = arcsin(x) = sin⁻¹(x)

Variables: x ∈ [−1, 1], θ ∈ [−90°, 90°]

Worked Examples

Example 1: Finding an angle of elevation

A ramp rises 2 m over a hypotenuse of 5 m. Find the angle.

Step 1:sin(θ) = 2/5 = 0.4
Step 2:θ = arcsin(0.4)

θ ≈ 23.58°

Common Mistakes & Tips

  • !Entering a value outside [−1, 1] — arcsin is only defined for inputs in this range.
  • !Confusing sin⁻¹(x) with 1/sin(x) — sin⁻¹ means the inverse function, not the reciprocal.
  • !Forgetting that the principal value is limited to [−90°, 90°] and other solutions may exist.

Related Concepts

Arccos Calculator

The inverse cosine function.

Arctan Calculator

The inverse tangent function.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arccos Calculator (Inverse Cosine)

Arctan Calculator (Inverse Tangent)

Sine Calculator

Trigonometric Functions Calculator

Frequently Asked Questions

What happens if I enter a value outside [-1, 1]?

The result is undefined (NaN) because no real angle has a sine greater than 1 or less than −1.

Are there other angles with the same sine?

Yes. If θ = arcsin(x), then 180° − θ also has the same sine. Additionally, θ + 360°n for any integer n.