# Inverse Trigonometric Functions Calculator

Created by Wei Bin Loo
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Jun 05, 2023

This inverse trigonometric functions calculator will help you solve all calculations involving the inverse trigonometric functions.

In this article, we'll help you understand what the inverse trigonometric functions are, what the inverse trigonometric functions ranges are, and how to calculate the inverse trigonometric functions. We will also demonstrate some practical examples to help you understand these concepts.

## What are the trigonometric functions and their inverses?

Trigonometric functions are mathematical functions that help to describe the relationship between the sides of a right triangle. The most common trigonometric functions are the sine, cosine and tangent, and more rarely you may come across the cotangent, secant and cosecant. You can learn more about this topic with our trigonometric functions calculator and trigonometry calculator.

The inverses of these functions are arcsine, arccosine, arctangent, arccotangent, arcsecant and arccosecant. The inverse trigonometric functions ranges and other useful information is summarized in the table below.

Function

Domain

Range

arcsin(x)

-1 ≤ x ≤ 1

-π/2 ≤ θ ≤ π/2

arccos(x)

-1 ≤ x ≤ 1

0 ≤ θ ≤ π

arctan(x)

All real numbers

-π/2 < θ < π/2

arccot(x)

All real numbers

0 < θ < π

arcsec(x)

x ≤ -1 or x ≥ 1

0 ≤ θ < π/2 or π/2 < θ ≤ π

arccsc(x)

x ≤ -1 or x ≥ 1

-π/2 ≤ θ < 0 or 0 < θ ≤ π/2

When applied to an angle, trigonometric functions return the ratio of the sides of a right triangle. So, in contrast, inverse trigonometric functions return the angle between two sides of a right triangle when they are applied to the ratio of these sides.

For instance, arcsin(x) returns the angle when applied to the ratio of the opposite side of the triangle to the hypotenuse, arccos(x) returns the respective angle when applied to the ratio of the adjacent side of the triangle to the hypotenuse and arctan(x) returns the angle when applied to the ratio of the opposite side of the triangle to the adjacent side.

Now, let's look at how to calculate the inverse trigonometric functions.

## How to calculate inverse trigonometric functions

Inverse trigonometric functions are used to calculate the angles in a right-angled triangle when the ratio of the sides adjacent to that angle is known.

To understand both concept and calculation, let's look at how to calculate the arcsine for the following right triangle.

• The inverse trigonometric function is arcsin (also denoted as sin-1).
• The domain of arcsin (the range of valid inputs) is -1 ≤ x ≤ 1 .
• The range of arcsin (the range of possible outputs) is 0 ≤ θ ≤ π.
• The opposite side's length is 2 cm for our example.
• The hypotenuse's length is 4 cm.

For this calculation, we would like to determine the angle (θ) with arcsin:

θ = arcsin(opposite / hypotenuse) = arcsin(2 / 4) = arcsin(1 / 2)

To calculate the inverse of the trigonometric function, use the formula:

sin(θ) = opposite / hypotenuse
==> sin(θ) = 2 / 4 = 1 / 2

We happen to know that sin(30°) is 1/2. Hence, θ = 30°.

You can also display the answer in radians, which will be 30 × (π / 180) = 0.5236 rad.

To understand more on this topic, please use our right triangle trigonometry calculator.

## FAQ

### What is the arcsin of 0.5?

The arcsin of 0.5 is 0.5236 radians or 30 degrees. The arcsin is the inverse of sin. Hence, if x = sin(θ), then θ = arcsin(x).

### What are the applications of inverse trigonometric functions?

Inverse trigonometric functions are used in various fields:

1. Engineering: Calculate angles in right-angled triangles when constructing a building.
2. Physics: Calculate the direction and speed of a wave.
3. Astronomy: Calculate the position of stars.
4. Mathematics: Solve complex equations.

### What is the range of the input for arccos?

The input for arccos can only be from -1 to 1. This is because the cos function can only produce values from -1 to 1.

### What are the different types of inverse trigonometric functions?

In total, there are 6 different types of inverse trigonometric functions. They are arcsine, arccosine, arctangent, arccotangent, arcsecant and arccosecant.

### How can I calculate the inverse trigonometric functions?

You can calculate the inverse trigonometric functions in three steps:

1. Identify the inverse trigonometric function you need to calculate.

2. Substitute θ into the appropriate inverse trigonometric function, and solve for the angle that satisfies the equation

3. Check that the angle you obtained is within the domain of the trigonometric function under consideration.

Wei Bin Loo
Inverse trigonometric function
arcsin
x
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