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PSLE

PSLE Foundation Mathematics

<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>

De Moivre's Theorem

Properties of Trigonometric and Exponential Forms

Cubed sine from a triple angle identity

Complex Numbers

<p>Use complex number methods to find the exact value of $\cos\dfrac{7\pi}{12}$ and $\sin\dfrac{7\pi}{12}$.</p>

Use complex number methods to find the exact value of $\cos\dfrac{7\pi}{12}$ and $\sin\dfrac{7\pi}{12}$.

Modulus, Argument, and Operations on Complex Numbers

Exact trigonometric ratios via the exponential form

<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>Given that $z={{e}^{\frac{3\pi }{4}\mathrm{i}}}$, find the smallest positive real number $n$ for ${{z}^{n}}$ to be positive real number.</p>

Given that $z={{e}^{\frac{3\pi }{4}\mathrm{i}}}$, find the smallest positive real number $n$ for ${{z}^{n}}$ to be positive real number.

Worked Example 5

<p>Given that $z={{e}^{-\frac{\pi }{6}\mathrm{i}}}$, find the smallest positive integer $n$ for ${{z}^{*}}{{z}^{n}}$ to be purely imaginary.</p>

Given that $z={{e}^{-\frac{\pi }{6}\mathrm{i}}}$, find the smallest positive integer $n$ for ${{z}^{*}}{{z}^{n}}$ to be purely imaginary.

Using Argument to Determine The Nature of a Complex Number Q3

<p>Given that $z={\mathrm{e}}^{-\frac{\pi}{3}\mathbf{i}}$, find the smallest positive integer $n$ for $z^{*}z^{2n}$ to be a negative real number.</p>

Given that $z={\mathrm{e}}^{-\frac{\pi}{3}\mathbf{i}}$, find the smallest positive integer $n$ for $z^{*}z^{2n}$ to be a negative real number.

Using Argument to Determine The Nature of a Complex Number Q2

<p>Given that $z={{e}^{\frac{3\pi }{4}\mathrm{i}}}$, find the smallest positive real number $n$ for ${{z}^{n}}$ to be a positive real number.</p>

Given that $z={{e}^{\frac{3\pi }{4}\mathrm{i}}}$, find the smallest positive real number $n$ for ${{z}^{n}}$ to be a positive real number.

Using Argument to Determine The Nature of a Complex Number Q1

<p>Express $2+2e^{i\theta}$ in the form $r{\mathrm{e}}^{\mathbf{i},\alpha}$ exactly, where $-\pi <\alpha \le \pi$.</p>

Express $2+2e^{i\theta}$ in the form $r{\mathrm{e}}^{\mathbf{i},\alpha}$ exactly, where $-\pi <\alpha \le \pi$.

Special Results involving Exponential Forms Q6

<p>Show that $e^{i\theta}-e^{-i\theta}$ is an imaginary number for all $\theta$.
The complex number $w$ is given by $w=\dfrac{2}{1-e^{i2\theta}}$, where $0<\theta<\pi$.
Show that $\mathrm{Im}(w)=\cot\theta$.</p>

Show that $e^{i\theta}-e^{-i\theta}$ is an imaginary number for all $\theta$.
The complex number $w$ is given by $w=\dfrac{2}{1-e^{i2\theta}}$, where $0<\theta<\pi$.
Show that $\mathrm{Im}(w)=\cot\theta$.

Special Results involving Exponential Forms Q5

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

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

Special Results involving Exponential Forms Q4

<p>Show that ${{e}^{\mathrm{i}\theta }}+1=\left( 2\cos \frac{\theta }{2} \right){{e}^{\mathrm{i}\frac{\theta }{2}}}$.</p>

Show that ${{e}^{\mathrm{i}\theta }}+1=\left( 2\cos \frac{\theta }{2} \right){{e}^{\mathrm{i}\frac{\theta }{2}}}$.

Special Results involving Exponential Forms Q3

<p>Show that $e^{\mathrm{i}\theta}+e^{-\mathrm{i}\theta}=2\cos \theta$.</p>

Show that $e^{\mathrm{i}\theta}+e^{-\mathrm{i}\theta}=2\cos \theta$.

Special Results involving Exponential Forms Q2

<p>Show that $e^{\mathrm{i}\theta}-e^{-\mathrm{i}\theta}=2\mathrm{i}\sin \theta$.</p>

Show that $e^{\mathrm{i}\theta}-e^{-\mathrm{i}\theta}=2\mathrm{i}\sin \theta$.

Special Results involving Exponential Forms Q1

<p>Let $z=a+\mathrm{i}b$ where $0 < a < \frac{\pi }{2}$ and $b$ is a real number. Evaluate the exact values of $a$ and $b$ such that ${{e}^{\mathrm{i}\left( z+\frac{\pi }{6} \right)}}=\sqrt{2}\cos \frac{\pi }{4}+\mathrm{i}2\sin \frac{\pi }{3}$.</p>

Let $z=a+\mathrm{i}b$ where $0 < a < \frac{\pi }{2}$ and $b$ is a real number. Evaluate the exact values of $a$ and $b$ such that ${{e}^{\mathrm{i}\left( z+\frac{\pi }{6} \right)}}=\sqrt{2}\cos \frac{\pi }{4}+\mathrm{i}2\sin \frac{\pi }{3}$.

Properties from Exponential or Trigonometric form Q9

<p>Given that $z=r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, solve for $z$ in the form $x+y\mathrm{i}$ if $4+\mathrm{i}\left( 4z+1 \right)=2r{{e}^{\mathrm{i}\left( \pi +\theta \right)}}$.</p>

Given that $z=r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, solve for $z$ in the form $x+y\mathrm{i}$ if $4+\mathrm{i}\left( 4z+1 \right)=2r{{e}^{\mathrm{i}\left( \pi +\theta \right)}}$.

Solving Complex Equations

Properties from Exponential or Trigonometric form Q8

<p>Given that $z=e^{i\theta}$, find the value of $\theta$ in radians if
$3+i(3z+1)=e^{i(\pi+\theta)}$ and $-\pi<\theta<\pi$.</p>

Given that $z=e^{i\theta}$, find the value of $\theta$ in radians if
$3+i(3z+1)=e^{i(\pi+\theta)}$ and $-\pi<\theta<\pi$.

Properties from Exponential or Trigonometric form Q6

<p>Without the use of a calculator, express $\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}$ and $1-\mathrm{i}$ in exponential form. Hence evaluate ${{\left( \frac{\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}}{1-\mathrm{i}} \right)}^{12}}$.</p>

Without the use of a calculator, express $\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}$ and $1-\mathrm{i}$ in exponential form. Hence evaluate ${{\left( \frac{\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}}{1-\mathrm{i}} \right)}^{12}}$.

Cartesian, Trigonometric and Exponential Forms

Properties from Exponential or Trigonometric form Q4

<p>It is given that</p>
<p style="text-align: center">$z=\cos \theta +\mathbf{i}\sin \theta $, $-\pi \le \theta \le \pi $.</p>
<p>Show clearly that</p>
<p style="text-align: center">$\frac{2}{1+z}=1-\mathbf{i}\tan \left( \frac{\theta }{2} \right)$.</p>

$z=\cos \theta +\mathbf{i}\sin \theta $, $-\pi \le \theta \le \pi $.

$\frac{2}{1+z}=1-\mathbf{i}\tan \left( \frac{\theta }{2} \right)$.

Complex Numbers Q1

<p>Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{{u}_{r}}}$ in terms of ${r}$, where&nbsp;${{{u}_{r}}}$ is the ${r}$th term of the series.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that the series is geometric and convergent. Hence, state the sum to infinity.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By expressing the sum to infinity in terms of ${\sin \theta }$ and ${\cos \theta }$, show that</div><div class="list-item-paragraph" style="text-align: center">${\cos \theta +\frac{1}{2}\cos 2\theta +\frac{1}{4}\cos 3\theta +…=\frac{4\cos \theta -2}{5-4\cos \theta }}$.<span class="inline-marks">[5]</span></div></li></ol>

Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$

Proving the progression

Properties of Complex Conjugates

Geometric Progression

Sum to infinity

Application of Sum to Infinity

2023 TGM P1 Q8

Sequence and Series

<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;${\left| {{\mathrm{e}}^{\mathrm{i}\theta }} \right|=1}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Consider the geometric series ${1+\frac{1}{3}{{\mathrm{e}}^{\mathrm{i}\theta }}+\frac{1}{9}{{\mathrm{e}}^{2\mathrm{i}\theta }}+…}$</div><div class="list-item-paragraph">Write down the common ratio, ${z}$, of the series and show that&nbsp;${\left| z \right|=\frac{1}{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, show that&nbsp;${\sin \theta +\frac{1}{3}\sin 2\theta +\frac{1}{9}\sin 3\theta +…=\frac{9\sin \theta }{10-6\cos \theta }}$.<span class="inline-marks">[8]</span></div></li></ol>

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