Tan Inverse Calculator

Created by Anna Szczepanek, PhD
Reviewed by Davide Borchia
Last updated: Jan 20, 2023

With our tan inverse calculator, you can discover what inverting the tangent function is all about and what issues have to be overcome in the process. We'll also show you an example of determining tan inverse without any calculations (so you don't need to panic searching for a calculator with tan inverse 5 minutes before an exam), just pure reasoning! Let's go!

What is tan inverse in math?

Tan inverse is just the inverse of the tangent function you've encountered in school in the context of right triangles. Simply speaking, you use the tan inverse when you need to "reverse" the action of the tangent function, that is, to find an angle that produced some specific value of tangent.

The tan inverse function is often called arcus tangent and its usual symbol is arctan. We have the formula:

arctan(tan x) = x

for x ∈ (-π/2, π/2).

What is the range of tan inverse?

You may wonder why we've said above that x ∈ (-π/2, π/2). The reason is that the tangent is not an injective function (it is many-to-one and not one-to-one), so it cannot be inverted as it is. First, we need to restrict our attention to some interval where the tangent is one-to-one. The interval that people choose most often is (-π/2, π/2). The consequence of this is the range of tan inverse is the same interval (-π/2, π/2).

So we're done with the range. It's only natural that now we'll discuss the domain of tan inverse function.

What is the domain of tan inverse?

The domain of tan inverse, on the other hand, is all real numbers, because the range of the tangent is all real number. In general, the domain of a function becomes the range of its inverse, and the range of a function becomes the domain of its inverse.

How to use this tan inverse calculator?

Using our tan inverse calculator couldn't be any easier! Just insert the value in the field x, and the answer will immediately appear in the field y. Since, as we've discussed above, the domain of tan inverse is the set of all real numbers, you don't need to worry about anything!

Inverting the tangent function is such an important math operation that we've built a whole collection of calculators with tan inverse! Make sure to visit:


How do I determine tan inverse of 1?

You can find the tan inverse of 1 without any calculations:

  1. Sketch a right triangle and pick one of the acute angles.
  2. Recall that the tan is the ratio of the opposite to adjacent side.
  3. The tan is equal to 1 when the opposite and adjacent sides have the same length! That is, your triangle is a half of a square which was cut along its diagonal.
  4. Before cutting, the angle was equal to 90°.
  5. Cutting halved the angle, so it measures 45°. This is the answer!
Anna Szczepanek, PhD
y = arctan(x)
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