Cos-1 Calculator

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

Welcome to Omni's cos-1 calculator! It is here to help you solve problems connected to the inverse of the cosine function. All you need to do is to input a number and grab the result! Wanna learn more about the role of cos-1 in math? We've prepared a brief explanation - scroll down!

What is cos-1 in math?

Beware - the notation cos-1 have two very different meanings:

  • cos-1(x) = 1/cos(x), i.e., the multiplicative inverse of cos(x); or
  • cos-1(x) = arccos(x), i.e., the inverse function of the cosine. In other words, we have the problem of determining the angle whose cosine equals x.

We assume that you have in mind the inverse cosine.

How do I invert the cosine function?

The inverse of the cosine function is most often denoted by arccos. It answers the question of what angle leads to a particular value of cosine:

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

Looks easy, isn't it? However, there's a small trap! Namely, you can only calculate arccos for numbers in the interval [-1, 1], because cosine assumes only values between -1 and 1.

Look at the graph of the cos-1 function: you can see its domain.

Inverse cosine graph

In the graph, we also see that the range of arccos is the interval [0, π] (in radians: it equals the range [0,180°] in degrees). This is because the cosine is a periodic function; in particular, it is not bijective (one-to-one). Before inverting, we must restrict it to an interval where it is one-to-one. Most often, people choose [0, π] as the domain of this restricted cosine, and then this interval becomes the range of the inverse cosine.

How to use this cos-1 calculator?

Our cos-1 calculator is very easy to operate! You enter the argument x, and the value of cos-1(x) appears in the blink of an eye. But please remember that you must not enter values that are not between -1 and 1!

Have you noticed our tool *converts between radians and degrees?. No need to search for any additional tools!

Happy with the cos-1 calculator? Omni features a whole collection of tools dealing with inverse trigonometric functions:

FAQ

Is the inverse cosine function antisymmetric?

No, the inverse cosine function is not antisymmetric. Note, for instance, that cos-1(1) = 0 and cos-1(-1) = π; that is, cos-1(1) and cos-1(-1) are not opposite numbers. The inverse sine function, on the other hand, is antisymmetric, i.e., it satisfies sin-1(-x) = -sin-1(x).

How do I calculate cos-1 of one half?

To determine the inverse cosine of ½:

  1. Sketch a right-angled triangle.
  2. Recall that the cosine is equal to the ratio of the adjacent side to the hypotenuse.
  3. So we need to determine an angle suc that the hypotenuse is twice as long as the adjacent side.
  4. Dig into your memory and recall that this angle must be equal to 60°.
  5. If your memory no longer retains any stuff taught in geometry class, we recommend using an online cos inverse calculator ;)
Anna Szczepanek, PhD
y = arccos(x) , −1 ≤ x ≤ 1
x
y
deg
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