Operations with Complex Numbers

00:00 6 marks

Consider the complex numbers \(z_1 = 3 \operatorname{cis}\left(\frac{\pi}{4}\right)\) and \(z_2 = 2 \operatorname{cis}\left(-\frac{\pi}{3}\right)\).

(a) Find \(z_1 z_2\) in the form \(r \operatorname{cis} \theta\). [2]
(b) Find \(\frac{z_1}{z_2}\) in the form \(r \operatorname{cis} \theta\). [2]
(c) Find the value of \((z_2)^6\). [2]

Hint Hint

Hint

For each part, perform operations on the moduli and angles separately. Remember that for multiplication you add the angles and for division you subtract the angles. Use De Moivre’s theorem for computing powers.

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Complete Answer

Part (a): \(6 \operatorname{cis}\left(-\frac{\pi}{12}\right)\); Part (b): \(\frac{3}{2} \operatorname{cis}\left(\frac{7\pi}{12}\right)\); Part (c): \(64 \operatorname{cis}(0)\) or simply 64.

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