\begin{frame} \frametitle{Inverse Trigonometric Functions} \begin{minipage}{.49\textwidth} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=0cm] \diagram{-1.5}{1.5}{-1.1}{1.3}{1} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=-1:1,samples=300] (\x,{sin(\x*90)}); \end{scope} \diagramannotatez \diagramannotatexx{-1/$-\frac{\pi}{2}$,1/$\frac{\pi}{2}$} \diagramannotatey{1,-1} \end{tikzpicture} }\\[.5ex] {\small $\sin x$ restricted to $[-\frac{\pi}{2},\frac{\pi}{2}]$} \end{center} \end{minipage} \begin{minipage}{.49\textwidth} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=0cm] \diagram{-.5}{2.5}{-1.1}{1.3}{1} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0:2,samples=300] (\x,{cos(\x*90)}); \end{scope} \diagramannotatez \diagramannotatexx{1/$\frac{\pi}{2}$,2/$\pi$} \diagramannotatey{1,-1} \end{tikzpicture} }\\[.5ex] {\small $\cos x$ restricted to $[0,\pi]$} \end{center} \end{minipage} \pause\bigskip From \quad $f^{-1}(y) = x \iff f(x) = y$ \quad we get:\pause \begin{talign} \sin^{-1}(y) = x \quad&\iff\quad \sin(x) = y \text{ and } -\frac{\pi}{2} \le x \le \frac{\pi}{2}\\ \mpause[1]{\cos^{-1}(y) = x} \quad&\mpause[1]{\iff\quad \cos(x) = y \text{ and } 0 \le x \le \pi} \end{talign} \pause\vspace{-2ex} \begin{block}{} The \emph{inverse sine function} $\sin^{-1}$ is also denoted by $\arcsin$.\\\pause The \emph{inverse cosine function} $\sin^{-1}$ is denoted by $\arccos$. \end{block} \end{frame}