Section IIWritten responseCX2CX1

Question 1 · 9 marks

Spec and skills
  • Spec CX2
  • Spec CX1
Let z1=r1(cos⁡θ1+isin⁡θ1)z_1 = r_1(\cos\theta_1 + i \sin\theta_1) and z2=r2(cos⁡θ2+isin⁡θ2)z_2 = r_2(\cos\theta_2 + i \sin\theta_2), where r1>0r_1 > 0, r2>0r_2 > 0 and θ1\theta_1, θ2\theta_2 are real.
(a)
Prove that z1z2=r1r2(cos⁡(θ1+θ2)+isin⁡(θ1+θ2))z_1 z_2 = r_1 r_2\big(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\big), and hence that ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2| and arg⁡(z1z2)=arg⁡(z1)+arg⁡(z2)\arg(z_1 z_2) = \arg(z_1) + \arg(z_2).
4 marks

Put the proof in order.

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