Calculate the inverse sine (arcsin) of a ratio value. Maps ratio values between -1 and 1 back to angles.
The Arcsin Calculator is an inverse trigonometric tool designed to calculate the angle corresponding to a given sine value. Arcsine (commonly written as arcsin or sin⁻¹) is the inverse of the sine function. Given an input ratio x, it solves for the angle θ such that sin(θ) = x. The result can be expressed in degrees or radians.
Because the output of the sine function is bounded between −1 and 1, the domain of the arcsine function is strictly limited to the interval [−1, 1]. If you enter a value outside this range (such as 1.5), the calculator will return a domain error. Additionally, since sine is a periodic function (meaning multiple angles can yield the identical sine value), the output of the arcsin function is restricted to its principal values between −90° and 90° (or −π/2 and π/2 radians) to ensure it behaves as a valid function.
Arcsine is used in geometry, navigation, and engineering to determine unknown angles when side lengths are known. For example, if you know the height of a ramp and its length (hypotenuse), arcsine calculates the angle of incline. In physics, it is used to calculate refractive angles via Snell's Law. The arcsin calculator provides precise angle calculations, eliminating manual trigonometry table lookups and ensuring accuracy.
Understanding Inverse Sine (arcsin / sin⁻¹) and the Unit Circle
In geometry, a trigonometric ratio relates the angles of a right triangle to its side lengths. When extended to all real coordinates, we use the **Unit Circle** (a circle with radius r = 1 centered at the origin).
- Sine (sin θ): represents the vertical projection coordinate (y-coordinate) of the intersection point.
- Cosine (cos θ): represents the horizontal projection coordinate (x-coordinate) of the intersection point.
- Tangent (tan θ): represents the ratio of vertical to horizontal change (y/x). Undefined when the cosine is 0 (90° and 270°).
- Pythagorean Identity: For any angle, the relationship sin²(θ) + cos²(θ) = 1 always holds.
Arcsin Domain & Range: The sine function is not one-to-one, so to construct its inverse, we restrict the domain to [-90°, 90°] ([-π/2, π/2] radians). Therefore, the output angle of arcsin is always within this range.
How it Works & Formula
Calculates the inverse sine, returning the angle whose sine is the given value.
Practical Examples
If opposite is 1m and hypotenuse is 2m, arcsin(0.5) = 30°.
Frequently Asked Questions
What is the domain of arcsin?
The input must be between -1 and 1 (inclusive).