Determine na forma trigonométrica
Números complexos: Infinito 12 A - Parte 3 Pág. 140 Ex. 46
Enunciado
Sendo $$\begin{array}{*{20}{c}}
{z = 2\operatorname{cis} \frac{\pi }{3}}&{\text{e}}&{w = 3\operatorname{cis} \frac{\pi }{2}}
\end{array}$$ determine na forma trigonométrica:
- $zw$
- $\frac{z}{w}$
- ${z^3}$
Resolução
$$\begin{array}{*{20}{c}}
{z = 2\operatorname{cis} \frac{\pi }{3}}&{\text{e}}&{w = 3\operatorname{cis} \frac{\pi }{2}}
\end{array}$$
- Ora,
$$\begin{array}{*{20}{l}}
{zw}& = &{\left( {2\operatorname{cis} \frac{\pi }{3}} \right) \times \left( {3\operatorname{cis} \frac{\pi }{2}} \right)} \\
{}& = &{\left( {2 \times 3} \right)\operatorname{cis} \left( {\frac{\pi }{3} + \frac{\pi }{2}} \right)} \\
{}& = &{6\operatorname{cis} \frac{{5\pi }}{6}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{\frac{z}{w}}& = &{\frac{{2\operatorname{cis} \frac{\pi }{3}}}{{3\operatorname{cis} \frac{\pi }{2}}}} \\
{}& = &{\frac{2}{3}\operatorname{cis} \left( {\frac{\pi }{3} – \frac{\pi }{2}} \right)} \\
{}& = &{\frac{2}{3}\operatorname{cis} \left( { – \frac{\pi }{6}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{z^3}}& = &{{{\left( {2\operatorname{cis} \frac{\pi }{3}} \right)}^3}} \\
{}& = &{{2^3}\operatorname{cis} \left( {3 \times \frac{\pi }{3}} \right)} \\
{}& = &{8\operatorname{cis} \pi }
\end{array}$$
