\begin{frame} \frametitle{Angles} Angles can be measured in \emph{degrees} ($\textdegree$) or in \emph{radians} ($\rad$): \begin{itemize} \pause \item $180\textdegree = \pi\rad$ \item<3-> $360\textdegree = 2\pi\rad$ is a full revolution \end{itemize} \begin{center} \begin{tikzpicture}[default,nodes={scale=.8}] \draw (0,0) circle (1cm); \draw[fill=clred!20] (0,0) to (30:1cm) arc (30:0:1cm) -- cycle; \draw[->] (.4cm,0) arc (0:30:.4cm); \node at (15:.6cm) {$\alpha$}; \draw[dotted] (180:1.2) -- node [at end,right] {$0\textdegree = 0\rad$} node [at start,left] {$180\textdegree = \pi\rad$} (0:1.2); \pause\pause\pause\pause\pause\pause \draw[dotted] (180+90:1.2) -- node [at end,above,align=center] {$90\textdegree = $\\$\mpause[1]{\pi/2 \rad}$} node [at start,below] {$\mpause[2]{270\textdegree =} \mpause[3]{3\pi/2 \rad}$} (90:1.2); \pause\pause\pause\pause \draw[dotted] (180+30:0) -- node [at end,right] {$30\textdegree = \mpause[1]{\pi/6 \rad}$} (30:1.45); \pause\pause \draw[dotted] (180+45:0) -- node [at end,right] {$45\textdegree = \mpause[1]{\pi/4 \rad}$} (45:1.6); \pause\pause \draw[dotted] (180+60:0) -- node [at end,right] {$60\textdegree = \mpause[1]{\pi/3 \rad}$} (60:1.8); \pause\pause \draw[dotted] (180+60:0) -- node [at end,left] {$120\textdegree = 2\pi/3 \rad$} (90+30:1.8); \draw[dotted] (180+45:0) -- node [at end,left] {$135\textdegree = 3\pi/4 \rad$} (90+45:1.6); \draw[dotted] (180+30:0) -- node [at end,left] {$150\textdegree = 5\pi/6 \rad$} (90+60:1.45); \end{tikzpicture} \end{center} \setcounter{beamerpauses}{4} \mpause[0]{ From $180\textdegree = \pi\rad$ we conclude \begin{align*} 1\textdegree = \frac{\pi}{180} \rad && \mpause[1]{ \text{ and } } && \mpause[1]{ x\textdegree = \frac{x\cdot\pi}{180} \rad } \\[1ex] \mpause[2]{1\rad = \left( \frac{180}{\pi} \right)\textdegree} && \mpause[3]{ \text{ and } } && \mpause[3]{ x\rad = \left( \frac{x\cdot 180}{\pi} \right)\textdegree } \end{align*} } \end{frame}