Calcule o produto na forma trigonométrica

Forma trigonométrica do produto:

Se ${z_1} = {\rho _1}\operatorname{cis} {\theta _1}$ e ${z_2} = {\rho _2}\operatorname{cis} {\theta _2}$ são dois complexos não nulos, então $${z_1}.{z_2} = {\rho _1}{\rho _2}\operatorname{cis} \left( {{\theta _1} + {\theta _2}} \right)$$

1. Ora,
   $$\begin{array}{*{20}{l}}
   {{z_1}.{z_2}}& = &{3\operatorname{cis} \left( { – \frac{\pi }{3}} \right) \times \sqrt 2 \operatorname{cis} \frac{\pi }{6}} \\
   {}& = &{3\sqrt 2 \operatorname{cis} \left( { – \frac{\pi }{3} + \frac{\pi }{6}} \right)} \\
   {}& = &{3\sqrt 2 \operatorname{cis} \left( { – \frac{\pi }{6}} \right)}
   \end{array}$$
2. Ora,
   $$\begin{array}{*{20}{l}}
   {{z_2}.{z_3}}& = &{\sqrt 2 \operatorname{cis} \frac{\pi }{6} \times \operatorname{cis} \frac{{5\pi }}{6}} \\
   {}& = &{\sqrt 2 \operatorname{cis} \left( {\frac{\pi }{6} + \frac{{5\pi }}{6}} \right)} \\
   {}& = &{\sqrt 2 \operatorname{cis} \pi }
   \end{array}$$
3. Ora,
   $$\begin{array}{*{20}{l}}
   {{z_1}.{z_2}.{z_3}}& = &{3\operatorname{cis} \left( { – \frac{\pi }{3}} \right) \times \sqrt 2 \operatorname{cis} \frac{\pi }{6} \times \operatorname{cis} \frac{{5\pi }}{6}} \\
   {}& = &{3\sqrt 2 \operatorname{cis} \left( { – \frac{\pi }{3} + \frac{\pi }{6} + \frac{{5\pi }}{6}} \right)} \\
   {}& = &{3\sqrt 2 \operatorname{cis} \frac{{2\pi }}{3}}
   \end{array}$$