Resources for Trig Graphs
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Questions
12
With Worked SolutionClick Here -
Video Tutorials
2
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HSC Questions
10
With Worked SolutionClick Here
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}](/media/dr4ogrvp/4776.png)