data-ad-format="horizontal">



 

arccos

The arccos function is the inverse of the cosine function.
It returns the angle whose cosine is a given number.
Try this Drag any vertex of the triangle and see how the angle C is calculated using the arccos() function.

For every trigonometry function, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. (On some calculators the arccos button may be labelled acos, or sometimes cos-1.) So the inverse of cos is arccos etc. When we see "arccos x", we understand it as "the angle whose cosine is x"

cos30 = 0.866 Means: The cosine of 30 degrees is 0.866
arccos 0.866 = 30 Means: The angle whose cosine is 0.866 is 30 degrees.
Use arccos when you know the cosine of an angle and want to know the actual angle.
See also Inverse functions - trigonometry

Example - using arccos to find an angle

In the above figure, click on 'reset'. We know the side lengths but need to find the measure of angle C.
We know that so we need to know the angle whose cosine is 0.866, or formally: Using a calculator we find arccos 0.866 is 30°.

Large and negative angles

Recall that we can apply trig functions to any angle, including large and negative angles. But when we consider the inverse function we run into a problem, because there are an infinite number of angles that have the same cosine. For example 45° and 360+45° would have the same cosine. For more on this see Inverse trigonometric functions.

To solve this problem, the range of inverse trig functions are limited in such a way that the inverse functions are one-to-one, that is, there is only one result for each input value.

Range and domain of arccos

Recall that the domain of a function is the set of allowable inputs to it. The range is the set of possible outputs.

For y = arccos x :

Range
Domain

By convention, the range of arccos is limited to 0 to +180°. So if you use a calculator to solve say arccos 0.55, out of the infinite number of possibilities it would return 56.63°, the one in the range of the function.

Things to try

  1. In the figure above, click 'reset' and 'hide details'.
  2. Adjust the triangle to a new size
  3. Using the arccos function calculate the value of angle C from the side lengths
  4. Click 'show details' to check the answer.
While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus