Resources for Argand Diagram
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Questions
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HSC Questions
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Argand Diagram Theory
![Argand Diagram - \begin{multicols}{2} For the complex number \(z\), the Cartesian form is \[ z=x+i y \] The mod-arg form is \[ z=r(\cos \theta+i \sin \theta) \] where \(\bmod z=|z|=r\)\\ $\begin{aligned} & r=\sqrt{x^2+y^2} \\ & \arg z=\theta \text { for } -\pi<\theta \leq \pi \text { and } \\ & \tan \theta=\frac{y}{x} \end{aligned}$\\ \columnbreak \begin{center} \resizebox{0.3\textwidth}{!}{ \begin{tikzpicture} \coordinate[label=left:$O$] (O) at (0,0); \coordinate (A) at (2,0); \coordinate (B) at (2,2); \draw[line width=1pt,-latex] (0,0) -- (2,0)node[below,midway]{$x$}--(3,0) node[right] {$\Large x$}; \draw[line width=1pt,-latex] (0,0) -- (0,2.5)node[above] {$\Large y$}; \draw[line width=1pt] (0,0) -- (2,0)--node[right]{$y$} (2,2)node[above]{$P(x,y)$}--(0,0) node[above left,midway] {$r$}; \pic [ draw=black, line width=1pt, angle radius=0.5cm, angle eccentricity=1.5,"$\LARGE \theta$"] {angle=A--O--B}; \end{tikzpicture} } %\includegraphics[width=0.3\textwidth]{35b1cfe8-76cb-4aa6-8f4a-841209040e19} \end{center} \end{multicols} \begin{multicols}{2} \textbf{Example 1}\\ Find the cartesian form of \(z=\sqrt{2}\left(\cos \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right)\) \\ \textbf{Example 1 solution}\\ $\begin{aligned} \frac{x}{\sqrt{2}} & =\cos \frac{\pi}{4} \\ x & =1 \\ \frac{y}{\sqrt{2}} & =\sin \frac{\pi}{4} \\ y & =1 \\ \therefore z & =1+i \end{aligned}$\\ \begin{center} \resizebox{0.3\textwidth}{!}{ \begin{tikzpicture} \coordinate[label=left:$O$] (O) at (0,0); \coordinate (A) at (2,0); \coordinate (B) at (2,2); \draw[line width=1pt,-latex] (0,0) -- (2,0)node[below,midway]{$x$}--(3,0) node[right] {$\Large x$}; \draw[line width=1pt,-latex] (0,0) -- (0,2.5)node[above] {$\Large y$}; \draw[line width=1pt] (0,0) -- (2,0)--node[right]{$y$} (2,2)node[above]{$P(x,y)$}--(0,0) node[above left,midway] {$\sqrt{2}$}; \pic [ draw=black, line width=1pt, angle radius=0.5cm, angle eccentricity=1.5,"$\LARGE \frac{\pi}{4}$"] {angle=A--O--B}; \end{tikzpicture} } %\includegraphics[width=0.3\textwidth]{cc30a4e7-af0c-497f-b974-d3a8d159197a} \end{center} \columnbreak \textbf{Example 2}\\ Find the mod-arg form of \(z=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} i\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \bmod z & =\sqrt{\left(-\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2} \\ & =1 \\ \arg z & =\tan ^{-1}\left(\frac{\sqrt{3}}{2} \div-\frac{1}{2}\right) \\ & =\tan ^{-1}(-\sqrt{3}) \\ & =\frac{2 \pi}{3} \\ \therefore z & =\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3} \end{aligned}$\\ \begin{center} \resizebox{0.35\textwidth}{!}{ \begin{tikzpicture} \coordinate[label=below:$O$] (O) at (0,0); \coordinate (A) at (-2,0); \coordinate (B) at (-2,2); \draw[line width=1pt,latex-latex] (-3,0)--(0,0) -- (-2,0)node[below,midway]{$-\frac{1}{2}$}--(2,0) node[right] {$\Large x$}; \draw[line width=1pt,-latex] (0,0) -- (0,2.5)node[above] {$\Large y$}; \draw[line width=1pt] (-2,0)--node[left]{$\frac{\sqrt{3}}{2}$} (-2,2)node[above]{$P(x,y)$}--(0,0) node[above right,midway] {$r$}; \pic [ draw=black, line width=1pt, angle radius=0.5cm, angle eccentricity=1.5,"$\LARGE \theta$"] {angle=B--O--A}; \end{tikzpicture} } %\includegraphics[width=0.3\textwidth]{44e97327-3bec-4ca1-9819-f2daa5f4560d} \end{center} \end{multicols}](/media/anvnnl0t/argand-diagram.png)
