e^{i\theta}=\cos\theta+i\sin\theta

\cos\theta in exponential form?

\dfrac{e^{i\theta}+e^{-i\theta}}{2}

\sin\theta in exponential form?

\dfrac{e^{i\theta}-e^{-i\theta}}{2i}

Euler's relation and polar formEdexcel International A Level Further Maths: Flashcards

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Euler's relation?

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Euler's relation?: $e^{i\theta}=\cos\theta+i\sin\theta$
$e^{i\pi}$ ?: $-1$
$e^{-i\theta}$ in terms of cos and sin?: $\cos\theta-i\sin\theta$
Exponential form of a complex number with modulus $r$ , argument $\theta$ ?: $re^{i\theta}$
$re^{i\alpha}\times se^{i\beta}$ ?: $rs\,e^{i(\alpha+\beta)}$
$\dfrac{re^{i\alpha}}{se^{i\beta}}$ ?: $\dfrac rs\,e^{i(\alpha-\beta)}$
$\cos\theta$ in exponential form?: $\dfrac{e^{i\theta}+e^{-i\theta}}{2}$
$\sin\theta$ in exponential form?: $\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$
If $z=e^{i\theta}$ , what is $z+\frac1z$ ?: $2\cos\theta$
If $z=e^{i\theta}$ , what is $z-\frac1z$ ?: $2i\sin\theta$
$-1+i\sqrt3$ in the form $re^{i\theta}$ ?: $2e^{2\pi i/3}$
$2e^{i\pi/3}$ in the form $a+ib$ ?: $1+i\sqrt3$
Which angle unit must be used in $e^{i\theta}$ ?: Radians.

Exam questions on Euler's relation and polar form

Let $z=2e^{i\pi/3}$ and $w=3e^{i\pi/6}$ .
Find $z+w$ in the form $a+ib$ , giving $a$ and $b$ as exact values.2 marks
Let $\theta$ be a real number and let $z=e^{i\theta}$ .
By expanding $\left(z+\dfrac1z\right)^2$ and using $z^2=e^{2i\theta}$ , show that $\cos^2\theta=\frac12(1+\cos2\theta)$ .2 marks
The complex number $z=-1+i\sqrt3$ .
Write $z$ in the form $re^{i\theta}$ , where $-\pi<\theta\le\pi$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).