Complex Numbers: Converting and Multiplying

00:00 5 marks

Let \(z_1 = 2 \operatorname{cis}\left(\frac{\pi}{3}\right)\) and \(z_2 = \sqrt{2} + \sqrt{2}i\).

(a) Express \(z_1\) in Cartesian form \(a+bi\). [2]
(b) Express \(z_2\) in modulus-argument form \(r \operatorname{cis}\theta\). [2]
(c) Find \(z_1z_2\) in modulus-argument form. [1]

Part a

3 minutes 2 marks

Question

Express \(z_1 = 2 \operatorname{cis}\left(\frac{\pi}{3}\right)\) in Cartesian form \(a+bi\). [2]

Hint Hint

Hint

Use \(\cos(\pi/3)\) and \(\sin(\pi/3)\) to convert to Cartesian form.

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Complete Answer

\(z_1 = 1 + i\sqrt{3}\)

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Part b

3 minutes 2 marks

Question

Express \(z_2 = \sqrt{2} + \sqrt{2}i\) in modulus-argument form \(r \operatorname{cis}\theta\). [2]

Hint Hint

Hint

Determine the modulus and then compute the argument using the arctan function.

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\(z_2 = 2 \operatorname{cis}\left(\frac{\pi}{4}\right)\)

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Part c

2 minutes 1 marks

Question

Find \(z_1z_2\) in modulus-argument form. [1]

Hint Hint

Hint

Multiply the moduli and add the arguments from parts (a) and (b).

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Complete Answer

\(z_1z_2 = 4 \operatorname{cis}\left(\frac{7\pi}{12}\right)\)

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