Arctangent

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

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

Properties of Arctangent

• Domain (x):
• Range (θ):

  To define the inverse function of a function, it must necessarily be bijective (or one-to-one). The tangent function is not injective in the set of reals. By convention, the codomain is restricted to the interval to make the tangent function bijective.

  That is, since none of the six trigonometric functions are one-to-one, they must be restricted to have inverse functions. Thus the domains of the trigonometric functions are restricted appropriately so that they become one-to-one functions and their inverse can be determined.

• Continuity: continuous for all x in domain.
• Increasing-decreasing behaviour: increasing for all x in domain.
• Symmetry: odd (arctan(−x) = − arctan(x)).
• The derivative of arccosine function:
• The integral of arccosine function:
• End behaviour:
  

Arctangent Table

The arctangent of the most common values is:

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:

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).

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.