# Inverse Cosine Calculator

Table of contents

What is the inverse cosine?How do I calculate the inverse cosine function?More than the inverse cosine calculatorFAQsThe inverse cosine calculator will quickly solve your doubts about this useful function. Keep reading this short article to learn:

- What is the
**inverse cosine**; - The
**domain and range**of the inverse cosine; - How to
**calculate the inverse cosine**;

And much more!

## What is the inverse cosine?

The **inverse cosine** is the **inverse function** of the **cosine**. Let's analyze these two concepts separately:

- An
**inverse function**is a function that**reverses the action of another function**. If we consider the function $y = f(x)$, we define the inverse function as $x = f^{-1}(y)$. As you can see, the inverse function takes as an argument the output of the original function and returns the original input. Inverting the function correspond, in graphical terms, to**reflect the function**with respect to the diagonal of the first quadrant. - The
**cosine**is a basic**trigonometric function**that corresponds to the**horizontal projection of the radius of a circle**at a certain angle $\alpha$.

Let's take a look at the **cosine function**.

The cosine is a periodic function, which means the values it assumes between $0\degree$ and $360\degree$ repeat cyclically. The periodicity of the function means that the cosine's domain is the entire x-axis. The values of the function are entirely comprised in the range $[-1,1]$.

These two pieces of information allow us to define the domain and range for the inverse cosine function:

- The
**domain**is $[-1,1]$; and - The
**range**is $[0\degree,360\degree]$.

In the picture below, you can see the inverse cosine graph:

The inverse cosine takes as its argument a dimensionless number and gives an angle as a result: these quantities are what you are going to find on the horizontal and vertical axis when defining the inverse cosine function.

## How do I calculate the inverse cosine function?

Calculating the inverse cosine function is no easy task: apart from a small set of neat values, using a calculator to find its value from any real number in the domain is often the only way to work your way through this function: that's where Omni's inverse cosine calculator comes in handy!

Anyway, here are the most important values of the inverse cosine:

- $\arccos(0) = 90\degree$;
- $\arccos(1/2) = 60\degree$;
- $\arccos(1/\sqrt{2}) = 45\degree$;
- $\arccos(\sqrt{3}/2) = 30\degree$; and
- $\arccos(1) = 0\degree$;

## More than the inverse cosine calculator

Maybe you want to introduce your students to the inverse cosine function under another name: we made these calculators for you:

- The arccos calculator;
- The cos inverse calculator; and
- The cos-1 calculator.

### What is the domain of the inverse cosine?

The domain of the inverse cosine is `[-1,1]`

. This interval corresponds to the range assumed by the cosine function, which has a minimum at `180°`

and a maximum at `0°`

.

### What is the inverse cosine of 1/√2?

The inverse cosine of `1/√2`

is `45°`

. To find this result, follow this reasoning:

- What is the angle that has cosine equal to
`1/√2`

? The answer is`45°`

. - Call the angle
`x`

, and the result y:`y = cos(x)`

. Remember that by definition, the inverse cosine is the function`x = arccos (y)`

. - We then substitute the values found before:

`cos(45°) = 1/√2`

And:

`arccos(1/√2) = 45°`

.

### How do I calculate the inverse cosine of 0?

To find the inverse cosine of `0`

, use a graphical approach:

- Remember that the inverse cosine is nothing but the reflected cosine function.
- Trace the graph of the cosine function between
`0°`

and`180°`

, and find the point where the function crosses the x-axis: you will find`cos(90°) = 0`

. - You found the result of the inverse cosine of
`90°`

: simply write`arccos(0) = 90°`

!