2023TrialShort answer

Q13 (a) (i)

2 marks

Question

Use De Moivre’s Theorem to solve the equation

z^3=4\sqrt{2}(1+i).

Try the question before revealing the answer.

Solution

z^3=4\sqrt{2}(1+i)

\begin{aligned} &=8\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)\\ \text{Let }z&=r(\cos\theta+i\sin\theta)\\ \text{then }z^3&=r^3(\cos3\theta+i\sin3\theta)\\ r^3(\cos3\theta+i\sin3\theta)&=8\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)\\ r^3&=8\qquad 3\theta=\frac\pi4+2k\pi,\quad k\in\mathbb Z\\ r&=2\qquad \theta=\frac{\pi+8k\pi}{12} \end{aligned}

\begin{aligned} \text{when }k=0,\quad z&=2\left(\cos\frac\pi{12}+i\sin\frac\pi{12}\right)\\ k=1,\quad z&=2\left(\cos\frac{9\pi}{12}+i\sin\frac{9\pi}{12}\right)\\ &=2\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)\\ k=2,\quad z&=2\left(\cos\frac{17\pi}{12}+i\sin\frac{17\pi}{12}\right)\\ &=2\left(\cos\left(-\frac{7\pi}{12}\right)+i\sin\left(-\frac{7\pi}{12}\right)\right) \end{aligned}

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Syllabus alignment

Complex Numbers

Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.
Geometric representation of a complex numberRepresent complex numbers in the complex plane and use modulus, argument and polar form with their identities.

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