NSW Y12 Maths - Advanced Graphs and Equations Trig Graphs

Resources for Trig Graphs

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Trig Graphs Theory

In sketching trigonometric functions it is important to determine the amplitude and the period of the function.\\  For \(y=A \sin n \pi\) the amplitude is \(A\) and the period \(=\dfrac{2 \pi}{n}\).\\  \begin{multicols}{2}  \textbf{Example 1}\\ Sketch the curve \(y=2 \sin \left(x+\dfrac{\pi}{2}\right)\) in the domain \(0 \leq x \leq 2 \pi\).\\  \textbf{Example 1 solution}\\ Amplitude \(=2\) period \(=\dfrac{2 \pi}{1}=2 \pi\)\\  $$\begin{array}{|l|l|l|l|l|l|} \hline \mathrm{x} & 0 & \dfrac{\pi}{2} & \pi & \dfrac{3 \pi}{2} & 2 \pi \\ \hline \mathrm{y} & 2 & 0 & -2 & 0 & 2 \\ \hline \end{array}$$  \begin{center} \begin{tikzpicture}[         declare function={a(\x)=2*cos(deg(x));}     ] \def \domain{0:2*pi+0.5}  \def \xmax{2*pi+0.5} \def \xmin{-1} \def \ymax{3} \def \ymin{-3} \def \xlabel{x} \def \ylabel{y}  \begin{axis}[         axis lines=middle,         axis line style={Stealth-Stealth,very thick},         grid=both, %major,         ylabel = $\ylabel$,         xlabel = $\xlabel$,         width=3in, height=2.5in,         ymin=\ymin, ymax=\ymax,         xmin=\xmin, xmax=\xmax,         minor x tick num=1,         minor y tick num=1,         axis line style = thick,         major tick style = thick,         minor tick style = thick,         xtick distance = 2,         xlabel style={right},         ytick distance = 2,         ylabel style={above},         x grid style={thin, opacity=0.8},         y grid style={thin, opacity=0.8},         axis on top=false,         xtick={pi/6,pi/3,pi/2,2*pi/3,5*pi/6,pi,7*pi/6,4*pi/3,3*pi/2,5*pi/3,11*pi/6,2*pi},         ytick={-3,-2,...,3},         xticklabels={$\text{}$,         $\frac{\pi}{3}$,         $\text{}$,         $\frac{2\pi}{3}$,         $\text{}$,         $\pi$,         $\text{}$,         $\frac{4\pi}{3}$,         $\text{}$,         $\frac{5\pi}{3}$,         $\text{}$,         $2\pi$},   % Labels for x ticks %       extra x ticks={0}, %        yticklabels={$-1$, $-\frac{1}{2}$, $0$, $\frac{1}{2}$, $1$},  % Labels for y ticks         extra x tick style={xticklabel style={anchor=north east}} %        enlargelimits=true,         clip=false,     ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax);   % Plot the sine function \addplot[name path=a, ultra thick, ->, samples=300, smooth, domain=\domain, red] {a(x)}  node [pos=0.9, left, red, font=\small] {}; \end{axis} \end{tikzpicture}  %\includegraphics[width=0.3\textwidth]{Picture11} \end{center}  \columnbreak \textbf{Example 2}\\ Sketch the curve \( y=\cos (2 x-\pi)\) in the domain \(-\dfrac{\pi}{2} \leqslant x \leqslant \dfrac{\pi}{2}\)\\  \textbf{Example 2 solution}\\ Amplitude \(=1 \quad\) period \(=\dfrac{2 \pi}{2}=\pi\)\\  $$\begin{array}{|c|c|c|c|c|c|} \hline x & -\dfrac{\pi}{2} & -\dfrac{\pi}{4} & \pi & \dfrac{\pi}{4} & \dfrac{\pi}{2} \\ \hline y & 1 & 0 & -1 & 0 & 1 \\ \hline \end{array}$$  \begin{center} \begin{tikzpicture}[         declare function={a(\x)=cos(deg(2*x-pi));}     ] \def \domain{-pi/2-0.2:pi/2+0.2}  \def \xmax{pi/2+0.3} \def \xmin{-pi/2-0.3} \def \ymax{2} \def \ymin{-2} \def \xlabel{x} \def \ylabel{y}  \begin{axis}[         axis lines=middle,         axis line style={Stealth-Stealth,very thick},         grid=both, %major,         ylabel = $\ylabel$,         xlabel = $\xlabel$,         width=3in, height=2.5in,         ymin=\ymin, ymax=\ymax,         xmin=\xmin, xmax=\xmax,         minor x tick num=1,         minor y tick num=1,         axis line style = thick,         major tick style = thick,         minor tick style = thick,         xtick distance = 2,         xlabel style={right},         ytick distance = 2,         ylabel style={above},         x grid style={thin, opacity=0.8},         y grid style={thin, opacity=0.8},         axis on top=false,         xtick={-7*pi/12,-pi/2,-5*pi/12,-pi/3,-pi/4,-pi/6,-pi/12,0,pi/12,pi/6,pi/4,pi/3,5*pi/12,pi/2,7*pi/12},         ytick={-3,-2,...,3},         xticklabels={         $\text{}$,         $-\frac{\pi}{2}$,         $\text{}$,         $-\frac{\pi}{3}$,         $\text{}$,         $-\frac{\pi}{6}$,         $\text{}$,         $\text{}$,         $\text{}$,         $\frac{\pi}{6}$,         $\text{}$,         $\frac{\pi}{3}$,         $\text{}$,         $\frac{\pi}{2}$},   % Labels for x ticks %       extra x ticks={0}, %        yticklabels={$-1$, $-\frac{1}{2}$, $0$, $\frac{1}{2}$, $1$},  % Labels for y ticks         extra x tick style={xticklabel style={anchor=north east}} %        enlargelimits=true,         clip=false,     ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax);   % Plot the sine function \addplot[name path=a, ultra thick, <->, samples=300, smooth, domain=\domain, red] {a(x)}  node [pos=0.9, left, red, font=\small] {}; \end{axis} \end{tikzpicture} %\includegraphics[width=0.3\textwidth]{Picture22} \end{center}

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Videos

Videos relating to Trig Graphs.

  • Trig Graphs - Video - Trigonometric Functions and Graphing: Amplitude, Period, Vertical and Horizontal Shifts

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  • Trig Graphs - Video - Trigonometric Equations

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