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<p>Consider the equation $z^3=c$, where&nbsp;$z\in\mathbb{C}$ and $c$ is a positive constant.</p>
<p>One of the solutions of the equation is $z=-2+2i\sqrt3$.</p>
<ol class="abc-list"><li data-list-item-id="p2-tza-q08-item01" data-list-item-path="p2-tza-q08-item01"><div class="list-item-paragraph">Find the value of $c$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item02" data-list-item-path="p2-tza-q08-item02"><div class="list-item-paragraph">Write down the other two solutions of $z^3=c$ giving your answers in the form $a+bi$, where&nbsp;$a,b\in\mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item03" data-list-item-path="p2-tza-q08-item03"><div class="list-item-paragraph">Find the solutions of $(iz)^3=c$ giving your answers in the form $a+bi$, where&nbsp;$a,b\in\mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item04" data-list-item-path="p2-tza-q08-item04"><div class="list-item-paragraph">The solutions of $(iz)^3=c$ can be represented on an Argand diagram by three points $A$, $B$ and $C$.</div><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

Consider the equation $z^3=c$, where $z\in\mathbb{C}$ and $c$ is a positive constant.

One of the solutions of the equation is $z=-2+2i\sqrt3$.

Write down the other two solutions of $z^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$.

Find the solutions of $(iz)^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$.

De Moivre's Theorem

Finding Roots of Complex Numbers

Representing Complex Numbers on an Argand Diagram

IB AA HL May 2026 P2 TZA Q8

Complex Numbers

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Use complex number methods to show that&nbsp;$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the exact value of $\sin^3\dfrac{5\pi}{12}$.</div></li></ol>

Properties of Trigonometric and Exponential Forms

Cubed sine from a triple angle identity

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Use complex number methods to show that&nbsp;$\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the exact value of $\tan\dfrac{\pi}{8}$.</div></li></ol>

Double angle tangent identity via complex numbers

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-1760350733187-ft0ymlzq4" data-list-item-path="item-1760350733187-ft0ymlzq4"><div class="list-item-paragraph">It is given that&nbsp;$w=1+\mathrm{i}$ and $z=-\sqrt{3}-\mathrm{i}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760350743890-loro32v9r" data-list-item-path="item-1760350733187-ft0ymlzq4/subitem-1760350743890-loro32v9r"><div class="list-item-paragraph">Find $\frac{{{w}^{3}}}{z}$ in the form $r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, where&nbsp;$r&gt;0$ and $-\pi &lt;\theta \le \pi $.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760350744572-egh1i0ozn" data-list-item-path="item-1760350733187-ft0ymlzq4/subitem-1760350744572-egh1i0ozn"><div class="list-item-paragraph">Find the smallest positive integer value of $n$, such that&nbsp;${{\left( \frac{{{w}^{3}}}{z} \right)}^{n}}$ is real.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760350764080-yskxvfuvi" data-list-item-path="item-1760350764080-yskxvfuvi"><div class="list-item-paragraph">The complex number $q$ is given by $\frac{{{\mathrm{e}}^{\mathrm{i}\theta }}}{{{\mathrm{e}}^{-\mathrm{i}\theta }}-\mathrm{i}}$, where&nbsp;$0 &lt; \theta &lt; \pi $.</div><div class="list-item-paragraph">Show that the real part of $q$ is $\frac{1}{2}\left( 1-2\sin \theta  \right)$.<span class="inline-marks">[4]</span></div></li></ol>

**Do not use a calculator in answering this question.**

Modulus, Argument, and Operations on Complex Numbers

Cartesian, Trigonometric and Exponential Forms

2022 JPJC MYE P2 Q6

<ol class="abc-list"><li data-list-item-id="item-1760257139205-71hk9ns32" data-list-item-path="item-1760257139205-71hk9ns32"><div class="list-item-paragraph">Write down the modulus and argument of the complex number $w$, where&nbsp;$w=1+\mathbf{i}\sqrt{3}$. Hence find the set of values of the positive integer $n$ for which ${{w}^{n}}$is real, and show that, for these values of $n$, the value of ${{w}^{n}}$ is either ${{2}^{n}}$or $-{{2}^{n}}$.</div><div class="list-item-paragraph">Express ${{w}^{100}}-{{\left( w^* \right)}^{100}}$ in the form $k\mathbf{i}$, where&nbsp;$k$ is real, giving the exact value of $k$ in non-trigonometrical form.</div></li></ol>

Properties of Complex Conjugates

N1993 P2 Q12

<p>Show that&nbsp;$\frac{1+\mathrm{i}\tan \theta }{1-\mathrm{i}\tan \theta }={{\mathrm{e}}^{\mathrm{i}\left( 2\theta \right)}}$ where&nbsp;$-\frac{\pi }{2}&lt;\theta &lt;\frac{\pi }{2}$.</p>
<p>Hence find the value of ${{\left( \frac{1-\mathrm{i}\sqrt{3}}{1+\mathrm{i}\sqrt{3}} \right)}^{11}}$, giving your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\alpha }}$ where&nbsp;$r&gt;0$ and $-\pi &lt;\alpha \le \pi $.<span class="inline-marks">[5]</span></p>

Show that $\frac{1+\mathrm{i}\tan \theta }{1-\mathrm{i}\tan \theta }={{\mathrm{e}}^{\mathrm{i}\left( 2\theta \right)}}$ where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

Hence find the value of ${{\left( \frac{1-\mathrm{i}\sqrt{3}}{1+\mathrm{i}\sqrt{3}} \right)}^{11}}$, giving your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\alpha }}$ where $r>0$ and $-\pi <\alpha \le \pi $.

2013 MJC P1 Q11

<ol class="abc-list"><li><div class="list-item-paragraph">It is given that $z=1+\sqrt{3}\,\mathbf{i}$ is a root of the equation $3{{z}^{3}}+a{{z}^{2}}+bz-8=0$, where $a$ and $b$ are real numbers. Find the exact values of $a$ and $b$ and hence solve the equation completely, giving all the roots in exact form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Using Euler’s formula that ${{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $, show that ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{4}}=\cos 4\theta +\mathbf{i}\sin 4\theta $.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using result shown in <strong>(i)</strong>, by comparing the real parts, show that $\cos 4\theta ={{\cos }^{4}}\theta -6{{\cos }^{2}}\theta {{\sin }^{2}}\theta +{{\sin }^{4}}\theta $.</div><div class="list-item-paragraph">Obtain an expression for $\sin 4\theta $ in terms of $\cos \theta $ and $\sin \theta $. Hence show that $\tan 4\theta =\frac{4\tan \theta -4{{\tan }^{3}}\theta }{1-6{{\tan }^{2}}\theta +{{\tan }^{4}}\theta }$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence, find the possible values of $\tan \theta $ given that $\tan 4\theta =4$, leaving your answers correct to $3$ significant figures.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Equating Real and Imaginary Parts

Applying the Conjugate Root Theorem

2021 SAJC P1 Q7

<p>For this question, you may use the result ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{n}}=(\cos n\theta +\mathbf{i}\sin n\theta )$.</p>
<p>Find the exact value(s) of $\theta $, where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$, given that</p>
<p style="text-align: center">${{\left( \cos \theta +\mathbf{i}\sin \theta  \right)}^{3}}+{{(\cos \theta +\mathbf{i}\sin \theta )}^{5}}=-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$.</p><p style="text-align: right;">[6]</p>

For this question, you may use the result ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{n}}=(\cos n\theta +\mathbf{i}\sin n\theta )$.

Find the exact value(s) of $\theta $, where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$, given that

${{\left( \cos \theta +\mathbf{i}\sin \theta  \right)}^{3}}+{{(\cos \theta +\mathbf{i}\sin \theta )}^{5}}=-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$.

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