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Products and quotients in polar form

Products and quotients in polar form

IB Year 6 | Grade 12

Two complex numbers are given by \(z_1=2\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)\) and \(z_2=\sqrt{2}\left(\cos\dfrac{\pi}{12}+i\sin\dfrac{\pi}{12}\right)\).

  1. Find \(z_1 z_2\) and \(\dfrac{z_1}{z_2}\) in polar form.
  2. Hence express \(\dfrac{z_1}{z_2}\) in Cartesian form.

Solution:

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Answer:(a) \(z_1z_2=2\sqrt2\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right),\ \dfrac{z_1}{z_2}=\sqrt2\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)\) (b) \(1+i\)

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