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arctan

The arctan function is the inverse of the tangent function.
It returns the angle whose tangent is a given number.
Try this Drag any vertex of the triangle and see how the angle C is calculated using the arctan() 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 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.
Use arctan when you know the tangent of an angle and want to know the actual angle.
See also Inverse functions - trigonometry

Example - using arctan 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 tangent is 0.577, or formally: Using a calculator we find arctan 0.577 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 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.

Range and domain of arctan

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 = arctan x :

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° .

Things to try

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

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus