<p>Express the following complex numbers in the form $z=r{{e}^{\mathrm{i}\theta }}$, where&nbsp;$r\ge 0$ and $-\pi &lt;\theta \le \pi $,</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{\cos \theta +\mathrm{i}\sin \theta }{\cos \theta -\mathrm{i}\sin \theta }$</div></li><li><div class="list-item-paragraph">${{e}^{2+\mathrm{i}\theta }}$</div></li><li><div class="list-item-paragraph">$\frac{1}{1+{{e}^{\mathrm{i}\theta }}}$</div></li><li><div class="list-item-paragraph">$1+\cos \left( 2\theta \right)-\mathrm{i sin}\left( 2\theta \right)$</div></li></ol>

Express the following complex numbers in the form $z=r{{e}^{\mathrm{i}\theta }}$, where $r\ge 0$ and $-\pi <\theta \le \pi $,

$\frac{\cos \theta +\mathrm{i}\sin \theta }{\cos \theta -\mathrm{i}\sin \theta }$

$1+\cos \left( 2\theta \right)-\mathrm{i sin}\left( 2\theta \right)$

Complex Numbers - General

Complex Numbers

International Baccalaureate Higher Level Mathematics

IB HL Math

Expressing Various Complex Numbers in Exponential Form

$\frac{\cos\theta + i\sin\theta}{\cos\theta - i\sin\theta}$
\[= \frac{e^{i\theta}}{\cos(-\theta) + i\sin(-\theta)} = \frac{e^{i\theta}}{e^{-i\theta}}\]
\[= e^{i\theta - (-i\theta)} = e^{i(2\theta)}\]

$e^{2+i\theta}$
\[= e^2 \cdot e^{i\theta}\]
\[= e^2 e^{i\theta}\]

$\frac{1}{1 + e^{i\theta}}$

Multiply numerator and denominator by $e^{-i\frac{\theta}{2}}$:
\[= \frac{e^{-i\frac{\theta}{2}}}{e^{-i\frac{\theta}{2}} + e^{i\frac{\theta}{2}}}\]

Note: $e^{-i\frac{\theta}{2}} + e^{i\frac{\theta}{2}} = \cos\frac{\theta}{2} - i\sin\frac{\theta}{2} + \cos\frac{\theta}{2} + i\sin\frac{\theta}{2} = 2\cos\frac{\theta}{2}$

\[= \frac{e^{-i\frac{\theta}{2}}}{2\cos\frac{\theta}{2}} = \left(\frac{1}{2}\sec\frac{\theta}{2}\right) e^{-i\frac{\theta}{2}}\]

$1 + \cos 2\theta - i\sin 2\theta$

Using $\cos 2A = 2\cos^2 A - 1$ and $\sin 2A = 2\cos A \sin A$:
\[= 1 + (2\cos^2\theta - 1) - i(2\cos\theta\sin\theta)\]
\[= 2\cos^2\theta - i(2\cos\theta\sin\theta)\]
\[= 2\cos\theta[\cos\theta - i\sin\theta]\]
\[= 2\cos\theta[\cos(-\theta) + i\sin(-\theta)]\]
\[= (2\cos\theta) e^{-i\theta}\]

Expressing Various Complex Numbers in Exponential Form

Expressing Various Complex Numbers in Exponential Form

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