# Arccos Calculator (Inverse Cosine)

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Feb 02, 2023

Welcome to the arccos calculator, a.k.a. the inverse cosine calculator. Thanks to our tool you can quickly find arccos - which is, surprisingly, the primary use of this calculator. However, for those of you eager to learn more, we've prepared a short article explaining what the inverse of a cosine is, accompanied by a table and an inverse cosine graph. Also, if you're a bit reluctant or confused, jump to the section about arccos applications to read what inverse cosine has in common with Physics, Chemistry, or even house construction and work ergonomics!

## What is the inverse of cosine (arccos)?

Arccos is the inverse of a trigonometric function- specifically, the inverse of the cosine function. However, as trigonometric functions are periodic, then, in a strict sense, they cannot be inverted. We can deal with that problem by choosing an interval in which the basic function is monotonic. You can pick many different ranges, but for cosine the common choice is [0,π]. This range is called the set of principal values.

Abbreviation

Definition

Domain of $\mathrm{arccos}(x)$ for real values

Range of usual principal values

$\mathrm{arccos}(x)$

$x=\cos(y)$

$-1

$0\leq y \leq \pi$ $0\degree \leq y \leq 180\degree$

$\mathrm{acos}(x)$

$\mathrm{cos}^{-1}(x)$

$\mathrm{arccos}(x)$ is the most commonly used notation, as $cos^{-1}(x)$ may be misleading - remember that inverse cosine is not the same as the reciprocal of the function (in other words, raising to the power $-1$):

$\cos^{-1}(x) \neq \frac{1}{\cos(x)}$

## Inverse cosine graph

A function $\mathrm{f}$ has an inverse function if, and only if, $\mathrm{f}$ is a one-to-one function. The whole cosine function is not one-to-one, since

$\cos(x) = \cos(x+2\pi n)\ \mathrm{for\ every\ integer\ }n$

What can we do then?

As stated in the previous paragraph, we need to restrict the domain of basic periodic cosine function. Thus, as cosine is always in the range $[-1,1]$, and we choose the domain, $[0,\pi]$, the inverse cosine function properties will be the reverse of this:

• Inverse cosine domain of $x$ for real result: $[-1,1]$

• Inverse cosine range of usual principal value: $[0,\pi]$

In the table below you will find the inverse cosine graph, as well as some commonly used $\mathrm{arccos}$ values:

$\boldsymbol{x}$

$\boldsymbol{\mathrm{\textbf{arccos}}(x)}$

Graph*

$\mathrm{deg}$

$\mathrm{rad}$

$-1$

$180\degree$

$\pi$

$-\sqrt{3}/2$

$150\degree$

$5\pi/6$

$-\sqrt{2}/2$

$135\degree$

$\3pi/4$

$-1/2$

$120\degree$

$2\pi/3$

$0$

$90\degree$

$\pi/2$

$1/2$

$60\degree$

$\pi/3$

$\sqrt{2}/2$

$45\degree$

$\pi/4$

$\sqrt{3}/2$

$30\degree$

$\pi/6$

$1$

$0\degree$

$0$

Wondering where this inverse cosine graph comes from? It's simply created by reflecting the graph of cos x through the line $y = x$ (don't forget about our domain restrictions!):

## Inverse cosine - why should I care? Some obscure arccos applications

You may think that arccos is another useless term from trigonometry, but we want to convince you this is not the case! The inverse cosine function is really helpful for many scientific and real-life problems (awesome, isn't it?):

I Science

Maths:

• 📐Solving the triangle, using the law of cosines. If you know the three sides of a triangle, and you'd like to find any of the triangle's angles, you'll need to use arccos: find everything about this law at our law of cosines calculator.
• ⛓Calculating the shape of a hanging chain: arccos describes the catenary curve: meet this wonderful mathematical creature at Omni's catenary curve calculator.

Physics:

• 🥢 *Finding the angle between two vectors/ lines / line segments - rearrange the dot product formula so that angle is the subject of equation. And there you are, an inverse cosine application.

• 💧 Calculating the hydraulic radius of a partially filled pipe is possible if you know wetted perimeter, calculated from a formula using arccos.

Chemistry:

• 🧪 Arccos is useful for estimating the optimal bond angles of polyatomic molecules, like e.g. $\mathrm{H_2O}$ or $\mathrm{CH_4}$.

II Real-life examples

• 🏠 Calculating your roof pitch, or inclination angle of the staircase (though, depending on what dimensions are given, the inverse sine calculator or inverse tangent calculator may also be handy)
• Designing a ramp for disabled people or pushchairs. Inverse cosine will be extremely useful if you know the ramp's length and the available horizontal distance.
• 🖥️ Even choosing an ergonomic position at work! If you want to properly set up your work station, you'll need to know your optimal desk height or standing desk height, but, as for positioning the monitor - finding the tilt angle or viewing angle is a way easier with this arccos calculator.

Are you convinced now? Don't wait any longer, use our inverse cosine calculator to solve (almost all) of your problems!

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