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