Represente na forma algébrica os números complexos
Números complexos: Infinito 12 A - Parte 3 Pág. 92 Ex. 52
Enunciado
Represente na forma algébrica os números complexos:
- $z = 5\operatorname{cis} \pi $
- $z = 3\operatorname{cis} \frac{\pi }{2}$
- $z = \sqrt 2 \operatorname{cis} \frac{{7\pi }}{4}$
- $z = \operatorname{cis} \frac{{7\pi }}{6}$
- $z = \sqrt 3 \operatorname{cis} \left( { – \frac{\pi }{3}} \right)$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
z& = &{5\operatorname{cis} \pi } \\
{}& = &{5\left( {\cos \pi + i\operatorname{sen} \pi } \right)} \\
{}& = &{5\left( { – 1 + 0i} \right)} \\
{}& = &{ – 5}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{3\operatorname{cis} \frac{\pi }{2}} \\
{}& = &{3\left( {\cos \frac{\pi }{2} + i\operatorname{sen} \frac{\pi }{2}} \right)} \\
{}& = &{3\left( {0 + i} \right)} \\
{}& = &{3i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\sqrt 2 \operatorname{cis} \frac{{7\pi }}{4}} \\
{}& = &{\sqrt 2 \left( {\cos \frac{{7\pi }}{4} + i\operatorname{sen} \frac{{7\pi }}{4}} \right)} \\
{}& = &{\sqrt 2 \left( {\cos \frac{\pi }{4} + i\operatorname{sen} \left( { – \frac{\pi }{4}} \right)} \right)} \\
{}& = &{\sqrt 2 \left( {\frac{{\sqrt 2 }}{2} – \frac{{\sqrt 2 }}{2}i} \right)} \\
{}& = &{1 – i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\operatorname{cis} \frac{{7\pi }}{4}} \\
{}& = &{\cos \frac{{7\pi }}{6} + i\operatorname{sen} \frac{{7\pi }}{6}} \\
{}& = &{ – \cos \frac{\pi }{6} – i\operatorname{sen} \frac{\pi }{6}} \\
{}& = &{ – \frac{{\sqrt 3 }}{2} – \frac{1}{2}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\sqrt 3 \operatorname{cis} \left( { – \frac{\pi }{3}} \right)} \\
{}& = &{\sqrt 3 \left( {\cos \left( { – \frac{\pi }{3}} \right) + i\operatorname{sen} \left( { – \frac{\pi }{3}} \right)} \right)} \\
{}& = &{\sqrt 3 \left( {\cos \frac{\pi }{3} – i\operatorname{sen} \frac{\pi }{3}} \right)} \\
{}& = &{\sqrt 3 \left( {\frac{1}{2} – \frac{{\sqrt 3 }}{2}i} \right)} \\
{}& = &{\frac{{\sqrt 3 }}{2} – \frac{3}{2}i}
\end{array}$$
