The tangent function is not a one-to-one function.

The usual choice for restricting the domain is \(\left (-\frac {\pi }{2},\frac {\pi }{2}\right )\).

We then revolve the graph around the \(y=\theta \) diagonal.

The vertical asymptotes become horizontal asymptote and represent end-behavior.

Characteristics

We can deduce many characteristics about arctangent from tangent.

\(\blacktriangleright \) The domain is all real numbers, \((-\infty , \infty )\).

\(\blacktriangleright \) The range is \(\left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )\)

\(\blacktriangleright \) Arctan has one zero and that is \(0\).

\(\blacktriangleright \) Arctan is an increasing function.

\(\blacktriangleright \) Arctan is a continuous function.

\(\blacktriangleright \) Arctan has no maximums or minimums.

\(\blacktriangleright \) End-Behavior

• \(\lim \limits _{x \to -\infty } \arctan (x) = -\frac {\pi }{2}\)
• \(\lim \limits _{x \to \infty } \arctan (x) = \frac {\pi }{2}\)

Coordinates

Given a point described with Cartesian (rectangular) coordinates, \((x, y)\), we can obtain polar coordinates via the equations

• \(r^2 = x^2 + y^2\)
• \(\theta = \arctan \left ( \frac {y}{x} \right )\)