\begin{frame}
  \frametitle{Derivatives of Trigonometric Functions}
  
  We investigate \alert{$\lim_{\phi\to 0}\frac{\sin \phi}{\phi}$}
  \pause\hfill \textcolor{gray}{(for simplicity assume $0 < \phi < \pi/2$)}

  \pause
  \begin{center}
    \begin{tikzpicture}[default,scale=.9]
      \coordinate (a) at (0,0);
      \coordinate (b) at (40:3);
      \coordinate (c) at (0:3);
      \coordinate (d) at ({cos(40)*3},0);
      \coordinate (e) at ($(c) + (0,{(tan 20)*3})$);
      \coordinate (f) at ($(c) + (0,{(tan 40)*3})$);
      
      \draw (a) -- node[above,yshift=.5mm] {$1$} (b);
      \draw (a) -- node[below] {$1$} (c);
      
      \begin{scope}[cgreen]
      \draw (b) arc (40:0:3);
      \node at (20:2.8) {$\phi$};
      \end{scope}
      \node[anchor=east] at (a) {$A$}; 
      \node[anchor=south,xshift=-1mm] at (b) {$B$}; 
      \node[anchor=north] at (c) {$C$};
      
      \mpause[1]{ 
      \begin{scope}[cred]
      \node[anchor=north] at (d) {$D$};
      \draw (b) -- node[pos=.65,left] {$\sin \phi$} (d);
      \end{scope}
      } 

      \mpause[4]{ 
      \begin{scope}[cblue]
      \draw (c) -- (e) -- node[anchor=west,at start] {$E$} (b);
      \end{scope}
      \begin{scope}[gray]
      \draw ($(c) + (-2mm,0)$) -- ++(0,2mm) -- ++(2mm,0);
      \draw ($(b) + (-140:2mm)$) -- ++(-50:2mm) -- ++(40:2mm);
      \end{scope}
      } 

      \mpause[6]{ 
      \begin{scope}[cblue]
      \draw (c) -- (f) -- node[anchor=west,at start] {$F$} (b);
      \end{scope}
      \begin{scope}[gray]
      \draw ($(b) + (-140:-2mm)$) -- ++(-50:2mm) -- ++(40:-2mm);
      \end{scope}
      } 

      \mpause[9]{
      \begin{scope}[xshift=60mm,yshift=10mm]
      \draw (0,0) -- node[below] {$a$} (3,0) --  node[right] {$b$} (3,1.5) -- cycle;
      \draw (26:1.2) arc (26:0:1.2);
      \node at (13:.9) {$\alpha$};
      \mpause[10]{ \node[align=center] at (1.5,-.7) {$\tan \alpha = \frac{b}{a}$}; }
      \mpause[11]{ \node[align=center] at (1.5,-1.2) {$b = a \cdot \tan \alpha$}; }
      \end{scope}
      }
    \end{tikzpicture}
  \end{center}\vspace{-2ex}
  
  \begin{talign}
    &\mpause[2]{\sin \phi = |BD| < \phi} 
    \mpause[3]{\quad\implies\quad \frac{\sin \phi}{\phi} < 1} 
    \\
    &
    \mpause[5]{ \phi < |CE| + |EB| } 
    \mpause[7]{\;\text{ \& }\; |EB| < |EF|}
    \mpause[8]{\quad\implies\quad \phi < |CE| + |EF| = |CF|} \\
    &\mpause[12]{\phi < |CF|} \mpause[13]{ = 1\cdot \tan \phi} \mpause[14]{= \frac{\sin \phi}{\cos \phi}} 
    \mpause[15]{\quad\implies\quad \alert{\cos \phi < \frac{\sin \phi}{\phi} < 1}} 
  \end{talign}%
  \pause[+19]%
  We use the Squeeze Theorem:\vspace{-1ex}
  \begin{talign}
    \lim_{\phi \to 0} \cos \phi = 1 = \lim_{\phi \to 0} 1 
    && \mpause[1]{\implies} && \mpause[1]{\alert{\lim_{\phi\to 0}\frac{\sin \phi}{\phi} = 1}}
  \end{talign}
\end{frame}