Let z_1 and z_2 be two complex numbers such that z_1=-iz_2 and (z_1z_2)=.

Question

Let $z_1$ and $z_2$ be two complex numbers such that $z_1=-iz_2$ and $\arg(z_1z_2)=\pi$. Then:

Accepted Answer

From $z_1=-iz_2$: $\arg(z_1)=\arg(z_2)-\dfrac\pi2$ (up to $2\pi$), i.e.
$$\arg(z_1)-\arg(z_2)=-\frac\pi2. \quad(i)$$

From $\arg(z_1z_2)=\pi$: $\arg(z_1)+\arg(z_2)=\pi$. $\quad(ii)$

Adding (i) and (ii): $2\arg(z_1)=\dfrac\pi2 \implies \arg(z_1)=\dfrac\pi4$, and then $\arg(z_2)=\dfrac{3\pi}4$.

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