# Sin-1 Calculator

Created by Anna Szczepanek, PhD
Reviewed by Davide Borchia
Last updated: Jan 21, 2023

We've built this sin-1 calculator to help you deal with inverting the sine function. Just input a number and get the result! And if you want to learn more about the sin-1 function and its properties, scroll down!

## What is sin-1 in math?

The expression sin-1 may mean two very different things:

• sin-1(x) = 1/sin(x), i.e., the multiplicative inverse of value sin(x); or
• sin-1(x) = arcsin(x), i.e., the inverse function of the sine. We're dealing here with the problem of determining the angle whose sine is equal to x.

We assume that you've come here with the latter meaning in mind. This is because the multiplicative inverse is very easy to determine by performing a simple division, while inverting the sine is quite non trivial.

## Inverting the sine function

The inverse of the sine function, denoted by arcsin, describes the angle that led to a particular sine value:

arcsin(x) = y if and only if x = sin(y).

However, we can only calculate arcsine for numbers from the interval [-1, 1]. This is because sine assumes values between -1 and 1.

As for the values assumed by arcsin(x), they fill in the interval in radians [-π/2, π/2] (equal to [-90°, 90°] in degrees). In other words, this interval constitutes the range of sin-1. This is because the sine function must be restricted before inverting to an interval where it is bijective (one-to-one), and the most popular choice is [-π/2, π/2].

And here's the graph of the sin-1 function: you can see both the domain and the range:

## How to use this sin-1 calculator?

Omni's sin-1 calculator is very user-friendly! Just enter the argument x and the value of sin-1(x) will appear immediately. Remember you must enter the value between -1 and 1!

Note that our tool can *convert between radians and degrees. You don't need to search for any additional tools!

Tip: one of the great features of Omni calculators is that they can work in reverse. And this sin-1 calculator can do that too! This means that you can use it as a sine calculator as well!

Here are Omni tools covering inverse trigonometric functions:

## FAQ

### What is sin-1 of zero?

The answer is zero, that is, sin-1(0) = 0. To arrive at this result, it's best to re-formulate the question: what is the angle that lies in the interval [-π/2, π/2] and its sine equals zero? A quick look at the graph of the sine function tells us that this angle is indeed zero, as we've claimed.

### How do I determine sin-1 of a negative number?

To determine the sine inverse of a negative number -x:

1. Take the absolute value of your number (i.e., remove the minus sign): x.
2. Compute the inverse sine of the value from Step 1: sin-1(x). Use an online inverse sine calculator if you need help.
3. Write the minus sign in front of the value obtained in Step 2: -sin-1(x).
4. This is your answer! We've used here the fact that sin-1 is an antisymmetric function: sin-1(-x) = -sin-1(x).
Anna Szczepanek, PhD
y = arcsin(x) , −1 ≤ x ≤ 1
x
y
deg
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