Arctan, also known as the inverse tangent function, is the mathematical operation that serves as the inverse of the tangent function. In other words, it “undoes” the effect of taking the tangent of an angle. The tangent function takes an angle as its input and returns the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle. Arctan, on the other hand, takes this ratio as its input and returns the angle whose tangent is equal to that ratio.

To help understand what arctan is the inverse of, let’s consider an example. Imagine you have a right triangle with an angle of 45 degrees. If you take the tangent of this angle, you would find that the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle is 1. In other words, tan(45 degrees) = 1.

Now, if you were to use arctan on this ratio of 1, it would return the angle whose tangent is 1. In this case, arctan(1) would equal 45 degrees. So, arctan “undoes” the effect of taking the tangent by returning the original angle.

It’s important to note that the tangent function has a periodic nature, meaning it repeats values at regular intervals. As a result, the arctan function is also periodic. To keep things standardized, the arctan function is typically defined with a principal range of -π/2 to π/2 (or -90 degrees to 90 degrees). This means that the output of arctan will always be within this range, representing angles in the first and fourth quadrants of the coordinate plane.

For example, if you take the tangent of an angle of 150 degrees, you would find that the ratio is -0.577. However, if you were to use arctan on this ratio, it would return an angle of -30 degrees, which is within the principal range of arctan.

Arctan is the inverse of the tangent function. It takes the ratio of the side lengths in a right triangle and returns the angle whose tangent is equal to that ratio. The principal range of arctan is typically defined as -π/2 to π/2, representing angles in the first and fourth quadrants.