Arcsin Calculator

The arcsin of a number tells you which angle has that number as its sine. Inputs run from -1 to 1, and the answer comes back as an angle, in degrees by default. You can switch the result unit to radians, gradians, or turns from the dropdown.

How arcsin works

Sine takes an angle and returns a ratio between -1 and 1. Arcsin runs that backward: feed it a ratio, get back the angle. $\sin(\theta) = y \implies \arcsin(y) = \theta$ . So if sin(30°) = 0.5, then arcsin(0.5) = 30°. You'll see it written as sin⁻¹(x) in textbooks or asin(x) in code, but it's the same function.

There's a catch. Sine isn't one-to-one: sin(30°) and sin(150°) both equal 0.5, and so do infinitely many other angles. To make the inverse behave like a real function, mathematicians restrict the output to one slice: angles between -90° and 90° (or -π/2 to π/2 in radians). That slice is called the principal value, and it's what every arcsin calculator returns.

How to use it

Type a value between -1 and 1 into the x field. The angle appears in the result field in your chosen unit. It works in reverse too: type an angle into the result field and the sine value fills back in.

Pick the angle unit from the dropdown next to the result. Degrees fit geometry and everyday problems; radians are what you want for calculus or anything calling Math.sin in code.

Where you'll actually use it

The classic case is solving a right triangle when you know the opposite side and the hypotenuse: arcsin(opposite ÷ hypotenuse) gives you the angle. Physics problems lean on it for projectile launch angles or working out how far a pendulum swings from rest. In graphics code it shows up when you need to recover an angle from a normalized direction vector, like figuring out where the sun is from the lighting on a surface.

Quick reference

• The input has to sit in [-1, 1]. Anything outside that has no real arcsin.

• The output always lands between -90° and 90° (or -π/2 and π/2 in radians).

• arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(1) = 90°, arcsin(-1) = -90°.

• Use radians for calculus and code; degrees for geometry and most practical work.

FAQ

Why does arcsin(2) return an error?

Sine never goes above 1 or below -1, so no angle has 2 as its sine. The function isn't defined outside [-1, 1].

Is sin⁻¹(x) the same as 1/sin(x)?

No, even though the notation looks like it should be. sin⁻¹(x) is the inverse function (arcsin); 1/sin(x) is the reciprocal, which is called cosecant or cosec(x).

Why is the answer always between -90° and 90°?

Sine repeats every 360° and hits every value in [-1, 1] infinitely many times. To turn it into a usable inverse, the range has to be restricted to a single 180° window. So even though sin(150°) also equals 0.5, arcsin(0.5) returns 30°, not 150°.