Arcsin Calculator (Inverse Sine)

Created by Luciano Mino
Last updated: Dec 16, 2022

With the arcsin calculator, you'll be able to find the inverse sine of any number easily.

In a few paragraphs, we will show:

  • What the arcsin or inverse sine function is;
  • How the arc sin formula relates to other trigonometric functions 📐; and
  • The inverse sine graph.

Let's dive right in!

What is arcsin, arc sine, or inverse sine?

After learning about the Pythagorean theorem, and trigonometric functions, the inverse trigonometric functions are the next step. They are needed, e.g., to find the angle x between vectors in the work formula: W = F × d × cos(x). See Omni Calculator's work calculator for more information about the equation and its variables. In this case, we would use the arc cos function.

Similarly, the arc sin function is the inverse of the sine function. When we calculate the arc sine of a number, we're trying to find the angle that, when we take its sine, is equal to that number.

For example, if we wanted to find the arcsin of √2/2, we need to ask:

Which angle, when we take its sine, is equal to √2/2?

Answer: 45°.

Therefore arcsin(√2/2) = 45°

The domain of the arc sine function is -1 ≤ x ≤ 1 and its range is -π/2 ≤ arcsin(x) ≤ π/2.

🙋 Feel free to experiment with our arcsin calculator/inverse sine calculator: just input any x value within this domain, and our tool automatically outputs the angle for you. Tip: the arcsin calculator lets you express the angle in any unit, such as radians or arcsec.

Inverse sine graph

By restricting ourselves to the domain we've previously defined, -1 ≤ x ≤ 1, we can plot the arc sine function to get the inverse sine graph.

Graph of the inverse sine function.

Arc sin formula from other trigonometric functions

Here's a list of useful relationships between the arc sin and other trigonometric functions:

  • sin(arcsin(x))=x\sin{(\arcsin{(x)})} = x.
  • cos(arcsin(x))=1x2\cos{(\arcsin{(x)})} = \sqrt{1 - x^{2}}.
  • tan(arcsin(x))=x1x2\tan{(\arcsin{(x)})} = \frac{x}{\sqrt{1 - x^{2}}}.
  • arcsin(x)=π2arccos(x)\arcsin{(x)} = \frac{\pi}{2} - \arccos{(x)}. Here, the arccos function is the inverse of the cosine function as well.
  • arcsin(x)=arcsin(x)\arcsin{(-x)} = -\arcsin{(x)}.

You can verify these expressions with the arcsin calculator 🔢 to ensure you understand the subject.

Luciano Mino
y = arcsin(x) , −1 ≤ x ≤ 1
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