Resources for Argand Diagram
-
Video Tutorials
1
Click Here
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)
NSW Y12 Maths - Extension 2 Complex Numbers Argand Diagram
Videos
Videos relating to Argand Diagram.
-
Argand Diagram - Video - Argand Diagram
Plans & Pricing
With all subscriptions, you will receive the below benefits and unlock all answers and fully worked solutions.
-
Teachers TutorsFeaturesFreeProAll ContentAll courses, all topicsQuestionsAnswersWorked SolutionsSystemYour own personal portalQuizbuilderClass ResultsStudent ResultsExam RevisionRevision by TopicPractise ExamsAnswersWorked SolutionsAwesome StudentsFeaturesFreeProContentAny course, any topicQuestionsAnswersWorked SolutionsSystemYour own personal portalBasic ResultsAnalyticsStudy RecommendationsExam RevisionRevision by TopicPractise ExamsAnswersWorked Solutions
Books / e-books
Course Book
Purchase the course book. This book either comes as a physical book or it can be purchased as an e-book.
Topic Books
You may choose to purchase the individual topic book from the main coursebook. These only come as e-books.
