2022TrialShort answer

Q16 (a) (ii)

2 marks

Question

Find the roots of $w^5=1$ . Express your answers in the form $\cos\theta+i\sin\theta$ , where $-\pi<\theta\le\pi$ .

Try the question before revealing the answer.

Solution

(\cos\theta+i\sin\theta)^5=1

\cos5\theta+i\sin5\theta=\cos2k\pi+i\sin2k\pi,\qquad \text{where }k\text{ is an integer}.

5\theta=2k\pi,\qquad \theta=\frac{2k\pi}{5}

w=\cos\frac{2k\pi}{5}+i\sin\frac{2k\pi}{5},\qquad k=0,\pm1,\pm2.

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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.

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