# Cos Inverse Calculator

The cos inverse calculator will help you deal with problems that require **inverting the cosine function**. It can't be any simpler: give us a number (between -1 and 1!), and we'll evaluate the cos inverse of this value. Why must the number be between -1 and 1? Read on to learn some theory behind the cos inverse, in particular, understand its **domain and range** or see how to compute the **cos inverse of negative values**.

## What is the cos inverse function?

The cos inverse is the inverse of the cosine function (no surprises here). That is, the cos inverse finds the angle that produces a particular cosine value. We denote this function by `arccos,`

and we have the following formula:

**arccos(x) = y if and only if x = cos(y)**

for **x ∈ [-1, 1]**.

This last condition describes the domain of cos inverse. We'll discuss it now and then move on to the range.

## What is the domain of cos inverse?

The domain of the cos inverse is **[-1, 1]**. In other words, you can calculate cos inverse only for values **between -1 and 1**. Why? Recall that the range of a function becomes the domain of its inverse. As the cosine has values between -1 and 1, this interval is the domain of the cos inverse.

## What is the range of cos inverse?

The range of cos inverse is the interval **[0, π]** in radians, or **[0,180°]** in degrees. Why? As you most certainly remember, the cosine is a periodic function; in particular, it is many-to-one. We cannot invert such functions! First, we must restrict it to an interval where the function is **one-to-one**. For the cosine, we most often restrict to the interval [0, π]. This is why this interval is the range of the cos inverse.

To better memorize the above results, take a look at the cos inverse graph:

## How to use this cos inverse calculator?

Our tool is extremely easy to use: **you input an argument x**, and the value of

`arccos(x)`

appears immediately.But please remember what you've learned about the domain of cos inverse! You must not enter values that do not satisfy the condition **-1 ≤ x ≤ 1**. Otherwise, the universe will explode. (Joking, our calculator will simply refuse your input, saving the universe. Try and see.)

## Similar Omni tools

Satisfied with the results returned by our cos inverse calculator? As inverse trig functions are quite tricky to deal with, we've built several tools around this topic:

- Arccos calculator;
- Cos-1 calculator;
- Inverse cosine calculator; and
- Inverse sine calculator.

## FAQ

### How do I calculate the cos inverse of negative numbers?

To compute the cos inverse of a negative number `-x`

:

- Determine the
**absolute value**of your number (i.e., remove the minus sign):`x`

. - Compute the
**cos inverse**of the value from Step 1:`arccos(x)`

. You may want to use an online cos inverse calculator to do that. **Subtract**the value obtained in Step 2 from`π`

, i.e., compute`π - arccos(x)`

.- This is your answer! We've used here the formula
`arccos(-x) = π - arccos(x)`

.

### What is the cos inverse of zero?

The answer is **90°**, that is, **π/2 rad**. One way to get this result is to reformulate the question and ask *what is the angle that lies in the interval [0, π] and its cosine is zero?* It may help to sketch the cosine graph (the usual wavy stuff) and check where it crosses the x-axis. A quick look at this graph will tell you that the angle we're looking for is indeed the correct angle.