IB Maths AA HL Ahl 1 13 Hl Polar and Euler Forms Questions

Practise converting between Cartesian, polar and Euler forms, multiplying or dividing moduli and arguments, and interpreting rotations or scaling on an Argand diagram.

Exam points

Convert a complex number between Cartesian, polar and Euler form, choosing an argument in the required interval and checking the representation.
Multiply or divide complex numbers in polar or Euler form by combining their moduli and arguments, then extract the requested form or component.
Interpret modulus and argument geometrically on an Argand diagram, including rotations, scaling and the position of products or quotients.

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Study AHL 1.13 (HL)—Polar and Euler forms conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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Let $z=1-\cos 2 \theta-\mathrm{i} \sin 2 \theta, z \in \mathbb{C}, 0 \leq \theta \leq \pi$ .

Find the modulus and argument of z in terms of $\theta$ . Express each answer in its simplest form.

EITHER

\begin{array}{ll} z=(1-\cos 2 \theta)-i \sin 2 \theta & \\ |z|=\sqrt{(1-\cos 2 \theta)^{2}+(\sin 2 \theta)^{2}} & \boldsymbol{M} \mathbf{1} \\ |z|=\sqrt{1-2 \cos 2 \theta+\cos ^{2} 2 \theta+\sin ^{2} 2 \theta} & \boldsymbol{A} \mathbf{1} \\ =\sqrt{2} \sqrt{(1-\cos 2 \theta)} & \boldsymbol{A} \mathbf{1} \\ =\sqrt{2\left(2 \sin ^{2} \theta\right)} & \\ =2 \sin \theta & \boldsymbol{A} \mathbf{1} \end{array}

let $\arg (z)=\alpha$ $\tan \alpha=-\frac{\sin 2 \theta}{1-\cos 2 \theta}$ $=\frac{-2 \sin \theta \cos \theta}{2 \sin ^{2} \theta}$ $=-\cot \theta$ $\arg (z)=\alpha=-\arctan \left(\tan \left(\frac{\pi}{2}-\theta\right)\right)$ $=\theta-\frac{\pi}{2}$
OR
$z=(1-\cos 2 \theta)-\mathrm{i} \sin 2 \theta$ $=2 \sin ^{2} \theta-2 i \sin \theta \cos \theta$ $=2 \sin \theta(\sin \theta-i \cos \theta)$ $=-2 i \sin \theta(\cos \theta+i \sin \theta)$ $=2 \sin \theta\left(\cos \left(\theta-\frac{\pi}{2}\right)+i \sin \left(\theta-\frac{\pi}{2}\right)\right)$ $|z|=2 \sin \theta$ $\arg (z)=\theta-\frac{\pi}{2}$

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IB Maths AA HL Ahl 1 13 Hl Polar and Euler Forms Questions question 1