\begin{frame} \frametitle{Trigonometric Functions: Inverse Tangent} \begin{minipage}{.4\textwidth} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=0cm] \diagram{-1.3}{1.3}{-3}{3}{1} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=-0.8:0.8,samples=300] (\x,{tan(\x*90)}); \end{scope} \diagramannotatez \diagramannotatexx{-1/$-\frac{\pi}{2}$,1/$\frac{\pi}{2}$} \diagramannotatey{1,-1} \begin{scope}[cred,dashed] \draw (-1,-3) -- (-1,3); \draw (1,-3) -- (1,3); \end{scope} \end{tikzpicture} }\\[.5ex] $\tan x$ restricted to $(-\frac{\pi}{2},\frac{\pi}{2})$ \end{center} \end{minipage}\pause \begin{minipage}{.59\textwidth} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=0cm] \diagram{-3}{3}{-1.3}{1.3}{1} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=-0.8:0.8,samples=300] ({tan(\x*90)},\x); \end{scope} \diagramannotatez \diagramannotatex{1,-1} \diagramannotateyy{-1/$-\frac{\pi}{2}$,1/$\frac{\pi}{2}$} \end{tikzpicture} }\\[.5ex] $\tan^{-1} x$ or $\arctan x$ \end{center} \end{minipage} \pause\medskip \begin{block}{} \begin{malign} \tan^{-1} y = x \;\;\iff\;\; \tan x = y \text{ and } -\frac{\pi}{2} < x < \frac{\pi}{2} \end{malign} \end{block} \pause The function $\arctan$ has domain \pause $(-\infty,\infty)$ and range \pause$(-\frac{\pi}{2},\frac{\pi}{2})$. \end{frame}