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The arctangent (notation: arctan or tan-1) is the inverse function of the tangent. That is:

Arctangent formula

As the arctangent and tangent are inverse functions, their composition is the identity. Thus:

Composition of arctangent and tangent

Properties of Arctangent

Arctangent Table

The arctangent of the most common values is:

Table of the arctangent of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)

Graphical Representation of the Arctangent Function

To better understand the graph of the arctangent, let’s first see the graphical representation of the tangent function:

Graph of the tangent function in arctangent function

As we see in the graph above the tangent is periodic, it is not one-to-one and the graph of the tangent function fails the horizontal line test. Hence the tangent does not have an inverse unless we restrict its domain. So, by convention, the domain of the tangent is usually restricted to the interval (-π/2, π/2).

Graph of the arctangent function

The graph of the arctangent function is symmetric to that of the tangent function to the bisector line of the first and third quadrants (y = x). With the restriction on the interval (-π/2, π/2) both functions are increasing and one is inverse of the other.

Graph of the tangent and arctangent functions, being symmetric with respect to the line y = x

AUTHOR: Bernat Requena Serra

YEAR: 2021


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