Represente na forma trigonométrica
Números complexos: Infinito 12 A - Parte 3 Pág. 142 Ex. 52
Enunciado
Sendo $$\begin{array}{*{20}{c}}
{z = \sqrt 2 – \sqrt 2 i}&{\text{e}}&{w = – \frac{2}{3} + \frac{2}{{\sqrt 3 }}i}
\end{array}$$ represente na forma trigonométrica.
- $z$
- $w$
- $zw$
- $\frac{z}{w}$
- ${w^3}$
- $\frac{1}{{ – w}}$
- ${z^2}\overline w $
- ${z^4}:{w^3}$
Resolução
$$\begin{array}{*{20}{c}}
{z = \sqrt 2 – \sqrt 2 i}&{\text{e}}&{w = – \frac{2}{3} + \frac{2}{{\sqrt 3 }}i}
\end{array}$$
- Ora,
$$\begin{array}{*{20}{l}}
z& = &{\sqrt 2 – \sqrt 2 i} \\
{}& = &{2\left( {\frac{{\sqrt 2 }}{{\text{2}}} – \frac{{\sqrt 2 }}{2}i} \right)} \\
{}& = &{2\operatorname{cis} \left( { – \frac{\pi }{4}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
w& = &{ – \frac{2}{3} + \frac{2}{{\sqrt 3 }}i} \\
{}& = &{\frac{4}{3}\left( { – \frac{1}{{\text{2}}} + \frac{{\sqrt 3 }}{2}i} \right)} \\
{}& = &{\frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{zw}& = &{2\operatorname{cis} \left( { – \frac{\pi }{4}} \right) \times \frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}} \\
{}& = &{\frac{8}{3}\operatorname{cis} \left( { – \frac{\pi }{4} + \frac{{2\pi }}{3}} \right)} \\
{}& = &{\frac{8}{3}\operatorname{cis} \frac{{5\pi }}{{12}}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{zw}& = &{\frac{{2\operatorname{cis} \left( { – \frac{\pi }{4}} \right)}}{{\frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}}}} \\
{}& = &{\frac{2}{{\frac{4}{3}}}\operatorname{cis} \left( { – \frac{\pi }{4} – \frac{{2\pi }}{3}} \right)} \\
{}& = &{\frac{3}{2}\operatorname{cis} \left( { – \frac{{11\pi }}{{12}}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{w^3}}& = &{{{\left( {\frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}} \right)}^3}} \\
{}& = &{{{\left( {\frac{4}{3}} \right)}^3}\operatorname{cis} \left( {3 \times \frac{{2\pi }}{3}} \right)} \\
{}& = &{\frac{{64}}{{27}}\operatorname{cis} \left( 0 \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{\frac{1}{{ – w}}}& = &{\frac{1}{{ – \frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}}}} \\
{}& = &{\frac{{\operatorname{cis} \left( 0 \right)}}{{\frac{4}{3}\operatorname{cis} \left( {\frac{{2\pi }}{3} + \pi } \right)}}} \\
{}& = &{\frac{1}{{\frac{4}{3}}}\operatorname{cis} \left( {0 – \frac{{2\pi }}{3} – \pi } \right)} \\
{}& = &{\frac{3}{4}\operatorname{cis} \left( { – \frac{{5\pi }}{3}} \right)} \\
{}& = &{\frac{3}{4}\operatorname{cis} \frac{\pi }{3}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{z^2}\overline w }& = &{{{\left( {2\operatorname{cis} \left( { – \frac{\pi }{4}} \right)} \right)}^2} \times \overline {\frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}} } \\
{}& = &{{2^2}\operatorname{cis} \left( { – 2 \times \frac{\pi }{4}} \right) \times \frac{4}{3}\operatorname{cis} \left( { – \frac{{2\pi }}{3}} \right)} \\
{}& = &{\frac{{16}}{{\frac{3}{3}}}\operatorname{cis} \left( { – \frac{\pi }{2} – \frac{{2\pi }}{3}} \right)} \\
{}& = &{\frac{{16}}{3}\operatorname{cis} \left( { – \frac{{7\pi }}{6}} \right)} \\
{}& = &{\frac{{16}}{3}\operatorname{cis} \frac{{5\pi }}{6}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{z^4}:{w^3}}& = &{\frac{{{{\left( {2\operatorname{cis} \left( { – \frac{\pi }{4}} \right)} \right)}^4}}}{{{{\left( {\frac{4}{3}\operatorname{cis} \frac{{2\pi }}{3}} \right)}^3}}}} \\
{}& = &{\frac{{16\operatorname{cis} \left( { – \pi } \right)}}{{\frac{{64}}{{27}}\operatorname{cis} 2\pi }}} \\
{}& = &{\frac{{16}}{{\frac{{64}}{{27}}}}\operatorname{cis} \left( { – \pi – 2\pi } \right)} \\
{}& = &{\frac{{27}}{4}\operatorname{cis} \left( { – 3\pi } \right)} \\
{}& = &{\frac{{27}}{4}\operatorname{cis} \pi }
\end{array}$$
