\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}