Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 6

(i)

If $z_1$ and $z_2$ are two complex numbers, show that:

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(i)

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).

z1z2=r1r2[cos⁡(θ1+θ2)+isin⁡(θ1+θ2)]\begin{aligned} z_1z_2 &=r_1r_2[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)] \end{aligned}

So:

Arg⁡(z1z2)=θ1+θ2=Arg⁡(z1)+Arg⁡(z2)\boxed{\operatorname{Arg}(z_1z_2)=\theta_1+\theta_2=\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)}

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