Product of complex numbers in trigonometric form

Product. $|z_1z_2|=2\cdot3=6$, $\arg(z_1z_2)=\pi/6+\pi/3=\pi/2$. So
$$z_1z_2=6(\cos\frac\pi2+i\sin\frac\pi2)=6i.$$

Quotient. $|z_1/z_2|=2/3$, $\arg(z_1/z_2)=\pi/6-\pi/3=-\pi/6$. So
$$\frac{z_1}{z_2}=\frac23\left(\cos\left(-\frac\pi6\right)+i\sin\left(-\frac\pi6\right)\right)=\frac23\left(\frac{\sqrt3}{2}-\frac i2\right)=\frac{\sqrt3}{3}-\frac i3.$$

Conclusion. $z_1z_2=6i$ and $z_1/z_2=\frac{\sqrt3}{3}-\frac{i}{3}$.