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\begin{frame}
  \frametitle{Trigonometric Functions: Identities}
  
  \begin{minipage}{.4\textwidth}
    \begin{tikzpicture}[default,scale=1.3]
      \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:.65cm) {$\alpha$};
      
      \onslide<3-6>{
        \draw[fill=clred!20] (0,0) to (-30:1cm) arc (-30:0:1cm) -- cycle;
        \draw[->] (.4cm,0) arc (0:-30:.4cm);
        \node at (-15:.7cm) {$-\alpha$};
      }
      \onslide<-2,7->{
        \draw[<->] (0,-.1) -- node[below] {{\tiny radius $1$}} (1,-.1);
      }

      \draw[dotted] (-1.2,0) -- (1.2,0);
      \draw[dotted] (0,-1.2) -- (0,1.2);
      
      \draw[dotted] (30:1cm) -- ({sqrt(3/4)},1.2);
      \draw[cgreen,<->] (0,1.2) -- node[above,inner sep=1mm] {$\cos \alpha$} ({sqrt(3/4)},1.2);
      \draw[dotted] (30:1cm) -- (1.2,1/2);
      \draw[cred,<->] (1.2,0) -- node[right,inner sep=1mm] {$\sin \alpha$} (1.2,1/2);
    \end{tikzpicture}
  \end{minipage}
  \begin{minipage}{.59\textwidth}
  {\small $\sin x$}\;\quad\scalebox{.5}{
  \begin{tikzpicture}[default,baseline=0cm]
    \diagram{-3}{7}{-1.1}{1.3}{1}
    \begin{scope}[ultra thick]
    \draw[cred] plot[smooth,domain=-3:7,samples=300] (\x,{sin(\x*90)});
    \end{scope}
    \diagramannotatez
    \diagramannotatexx{-2/$-\pi$,-1/$-\frac{\pi}{2}$,1/$\frac{\pi}{2}$,2/$\pi$,3/$\frac{3\pi}{2}$,4/$2\pi$,5/$\frac{5\pi}{2}$,6/$3\pi$}
    \diagramannotatey{1,-1}
  \end{tikzpicture}\hspace{1cm}
  }\\\medskip
  
  {\small $\cos x$}\quad\scalebox{.5}{
  \begin{tikzpicture}[default,baseline=0cm]
    \diagram{-3}{7}{-1.1}{1.3}{1}
    \begin{scope}[ultra thick]
    \draw[cgreen] plot[smooth,domain=-3:7,samples=300] (\x,{cos(\x*90)});
    \end{scope}
    \diagramannotatez
    \diagramannotatexx{-2/$-\pi$,-1/$-\frac{\pi}{2}$,1/$\frac{\pi}{2}$,2/$\pi$,3/$\frac{3\pi}{2}$,4/$2\pi$,5/$\frac{5\pi}{2}$,6/$3\pi$}
    \diagramannotatey{1,-1}
  \end{tikzpicture}\hspace{1cm}
  }
  \end{minipage}
  \pause
    
  Important identities:
  \begin{itemize}
  \pause
    \item $\sin (-\alpha) = \pause - \sin \alpha$
  \pause
    \quad and \quad $\cos (-\alpha) = \pause \cos \alpha$
  \pause
    \item $\sin (\alpha + 2\pi) = \sin \alpha$ \pause \quad and \quad $\cos (\alpha + 2\pi) = \cos \alpha$
  \pause
    \item $\cos \alpha = \sin (\alpha \alt<-9>{\alert{\pm ?}}{+ \frac{\pi}{2}})$
  \pause\pause
    \item $\sin^2 \alpha + \cos^2 \alpha =\pause 1$ \textcolor{gray}{(follows form the Pythagorean theorem)}
  \end{itemize}
  \pause\medskip

  \begin{indentation}{-.7cm}{-1cm}  
  \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
    \hline 
    $\alpha$ & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\frac{3\pi}{4}$ & $\frac{5\pi}{6}$ & $\pi$ & $\frac{3\pi}{2}$ & $2\pi$
    \\
    \hline 
    $\sin \alpha$ & $0$ & $\frac{1}{2}$ & $\frac{1}{\sqrt{2}}$ & $\mpause[1]{\frac{\sqrt{3}}{2}}$ & $\mpause[3]{1}$ & $\mpause[5]{\frac{\sqrt{3}}{2}}$ & $\mpause[5]{\frac{1}{\sqrt{2}}}$ & $\mpause[5]{\frac{1}{2}}$ & $0$ & $-1$ & $0$   
    \\
    \hline 
    $\cos \alpha$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{\sqrt{2}}$ & $\mpause[2]{\frac{1}{2}}$ & $\mpause[4]{0}$ & $\mpause[5]{-\frac{1}{2}}$ & $\mpause[5]{-\frac{1}{\sqrt{2}}}$ & $\mpause[5]{-\frac{\sqrt{3}}{2}}$ & $-1$ & $0$ & $1$
    \\
    \hline 
  \end{tabular}
  \end{indentation}
  \vspace{10cm}
\end{frame}