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 arctan button may be labelled atan, or sometimes tan-1.) So the inverse of tan is arctan etc. When we see "arctan x", we understand it as "the angle whose tangent is x"
| tan 30 = 0.577 | Means: The tangent of 30 degrees is 0.577 |
| arctan 0.577 = 30 | Means: The angle whose tangent is 0.577 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 tangent is 0.577, or formally:
Using a calculator we find arctan 0.577 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 tangent. For example 45° and 360+45° would have the same tangent. 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 | All real numbers |
By convention, the range of arctan is limited to -90° to +90° *.
So if you use a calculator to solve say arctan 0.55, out of the infinite number of possibilities it would return 28.81°, the one in the range of the function.
* Actually, -90° and +90° are themselves not in the range. This is because the tan function has a value of infinity at those values. But the values just below them are in the range, for example +89.9999999. But for simplicity of explanation, we say the range is ±90° .