# Arcsin Calculator (Inverse Sine)

By Hanna Pamuła, PhD candidate
Last updated: Mar 12, 2021

With this arcsin calculator (or inverse sine calculator) you'll have no issue finding the arcsine in your problem. Simply input the value of sine for the triangle and the angle in question will appear. The only thing you need to remember is the restricted domain of arcsine (−1 ≤ sine ≤ 1). If you're wondering what the arcsine is or what the graph of arcsin x looks like, wait no longer - scroll down and you'll find the answers below! We have also included a short paragraph about arcsine relationships, such as the relationship between arcsines integral and derivative. So, what are you waiting for?

## What is arcsine?

Arcsine is an inverse of the sine function. In other words, it helps to find the angle of a triangle which has a know value of sine. As sine's codomain for real numbers is [−1, 1] , we can only calculate arcsine for numbers in that interval.

Sine is a periodic function, so there are multiple numbers that have the same sine value. For example sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, sin(-π) = 0 and sin(-326π) = 0. Therefore, if somebody wants to calculate arcsin(0), the answer can be 0, 2π (360°) or -π (-180°), to name a few options! All of them are correct, but we usually only give one number called the principal value.

Abbreviation Definition Domain of arcsin x
for real result
Range of usual
principal values
arcsin(x)
sin-1x,
asin
x = sin(y) -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2
-90° ≤ y ≤ 90°

Arcsin(x) is the most common notation, as sin-1x may lead to confusion (because sin-1x ≠ 1/sin(x) ). The abbreviation asin in usually used in computer programming languages.

## Graph of arcsin x

As the basic function sine is not one-to-one, its domain must be restricted in order to ensure that arcsine is also a function. Usually, the chosen domain is -π/2 ≤ y ≤ π/2. This means that the range of the inverse function will be equal to the range of a principal function; thus the range of the arcsin function is [−π/2,π/2] and arcsine domain is between [−1,1]. Below you can find the graph of arcsin(x), as well as some commonly used arcsine values:

x arcsin(x) Graph
-1 -90° -π/2 -√3 / 2 -60° -π/3
-√2 / 2 -45° -π/4
-1/2 -30° -π/6
0 0
1/2 30° π/6
√2 / 2 45° π/4
√3 / 2 60° π/3
1 90° π/2

Wondering where this graph of arcsin x comes from? It may be found by reflecting a graph of sin(x), between the range of [-π/2 π/2], through the line y = x: ## Inverse sine, trigonometric functions, and other relationships

The relationships between the trigonometric functions and arcsine may help you understand the topic even better. A right-angled triangle that has a hypotenuse of length 1 is a good starting point. Just a quick reminder: for right-angled triangle, the sine function takes the angle θ and returns the ratio of the opposite/hypotenuse, which is equal to x in our exemplary triangle. The inverse sine function, arcsine, will take the ratio of the opposite/hypotenuse (x) and return the angle, θ. So, knowing that, for our triangle, arcsin(x) = θ we can also write that:

• Sine: sin(arcsin(x)) = x
• Cosine: cos(arcsin(x)) = √(1-x²)
• Tangent: tan(arcsin(x)) = x / √(1-x²)

Other useful relationships with arcsine are:

• arcsin(x) = π/2 - arccos(x)
• arcsin(-x) = -arcsin(x)

Sometimes the integral and derivative of arcsin are needed as well:

• integral of arcsin: ∫arcsin(x) dx = x arcsin(x) + √(1 - x²) + C
• derivative of arcsin: d/dx arcsin(x) = 1 / √(1 - x²) where x ≠ -1, 1

## Example of how to use the arcsin calculator Arcsine is a function which is useful e.g. in finding the angle of a right triangle. If you're searching for the angles in a right triangle and you know the side lengths, the well-known Pythagorean theorem wouldn't be so helpful. Finding the angles of a right triangle requires applying arcsine:

• for α: sin(α) = a / c so α = arcsin(a / c)
• for β: sin(β) = b / c so β = arcsin(b / c)

So let's assume we have two values given in a right triangle, a = 6 and c = 10, and we'd like to find the value of the angle α:

1. Input the value you want to find the arcsine of. In our case, it's 6/10. So, you can enter the value as 0.6, but the form 6/10 will also work. Just remember that the value should be between −1 and 1.
2. And... that's it! The arcsin calculator did its job and you've found the arcsine of your value. Now you know that arcsine(6/10) = 36.87°

Great! Now that you understand what arcsine is, maybe you'd like to check out more advanced applications of trigonometry? For instance, the law of sines (closely related to the law of cosines) is a must when solving triangle problems.

Hanna Pamuła, PhD candidate
y = arcsin(x) , −1 ≤ x ≤ 1
x
y
deg
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