Precalculus Review

Inverse tangent function

The inverse cosine of

, denoted by $\tan^{-1} u$ , is defined as the unique value $-\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}$ satisfying $\tan\theta = u$ . It is also known as the arctangent function $\arctan x$ . The domain and range are given by $D = [-\infty,\:\infty]$ and $R = \left[-\dfrac{\pi}{2},\:\dfrac{\pi}{2}\right]$ .

$\includegraphics{lec14c.ps}$

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}\qua... ...2 \pi + \tan^{-1} u & \quad\mbox{ if $u < 0$. } \end{array} \end{displaymath}$

Note that if

then $\tan^{-1} u < 0$ is out of the interval $[0,2\pi)$ .

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