Inverse tangent function

The inverse cosine of $ u$, denoted by  $ \tan^{-1} u$, is defined as the unique value $ -\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}$ satisfying $ \tan\theta = u$. The inverse tangent function `` $ \tan^{-1} x$'' is also known as the arctangent function ``$ \arctan x$.''

\includegraphics{lec14c.ps}

Domain and range of $ \tan^{-1}$. $ D = [-\infty,\:\infty]$ and $ R = \left[-\dfrac{\pi}{2},\:\dfrac{\pi}{2}\right]$.

Trigonometric equation. Consider the solutions to the equation  $ {\tan\theta = u}$ in the interval $ [0,2\pi)$ (which is the two cycle for the tangent function). The two solutions $ \theta_1$ and $ \theta_2$ are given by

\begin{displaymath}
\begin{array}{lll}
\theta_1 = \tan^{-1} u & \mbox{and}\quad
...
...} u
& \quad\mbox{ if $u < 0$. $^{\dagger\dagger}$ }
\end{array}\end{displaymath}

$ \dagger\dagger$ Note that if $ u < 0$ we find that $ \sin^{-1} u < 0$ and $ \tan^{-1} u < 0$ are out of the interval $ [0,2\pi)$.



Department of Mathematics
Last modified: 2005-09-29