Escreva $z$ na forma algébrica
Números complexos: Infinito 12 A - Parte 3 Pág. 141 Ex. 48
Enunciado
Escreva $z$ na forma algébrica:
- $z = \operatorname{cis} \frac{\pi }{3}$
- $z = 2\operatorname{cis} \left( { – \frac{\pi }{3}} \right)$
- $z = \sqrt 3 \operatorname{cis} \left( { – \frac{{5\pi }}{6}} \right)$
- $z = 2\operatorname{cis} \left( {\frac{{3\pi }}{4}} \right)$
- $z = \operatorname{cis} \frac{{9\pi }}{2}$
- $z = 9\operatorname{cis} 2\pi $
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
z& = &{\operatorname{cis} \frac{\pi }{3}} \\
{}& = &{\cos \frac{\pi }{3} + i\operatorname{sen} \frac{\pi }{3}} \\
{}& = &{\frac{1}{2} + \frac{{\sqrt 3 }}{2}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{2\operatorname{cis} \left( { – \frac{\pi }{3}} \right)} \\
{}& = &{2\left( {\cos \left( { – \frac{\pi }{3}} \right) + i\operatorname{sen} \left( { – \frac{\pi }{3}} \right)} \right)} \\
{}& = &{2\left( {\cos \frac{\pi }{3} – i\operatorname{sen} \frac{\pi }{3}} \right)} \\
{}& = &{2\left( {\frac{1}{2} – \frac{{\sqrt 3 }}{2}i} \right)} \\
{}& = &{1 – \sqrt 3 i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\sqrt 3 \operatorname{cis} \left( { – \frac{{5\pi }}{6}} \right)} \\
{}& = &{\sqrt 3 \left( {\cos \left( { – \frac{{5\pi }}{6}} \right) + i\operatorname{sen} \left( { – \frac{{5\pi }}{6}} \right)} \right)} \\
{}& = &{\sqrt 3 \left( { – \cos \frac{\pi }{6} – i\operatorname{sen} \frac{\pi }{6}} \right)} \\
{}& = &{\sqrt 3 \left( { – \frac{{\sqrt 3 }}{2} – \frac{1}{2}i} \right)} \\
{}& = &{ – \frac{3}{2} – \frac{{\sqrt 3 }}{2}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{2\operatorname{cis} \left( {\frac{{3\pi }}{4}} \right)} \\
{}& = &{2\left( {\cos \left( {\frac{{3\pi }}{4}} \right) + i\operatorname{sen} \left( {\frac{{3\pi }}{4}} \right)} \right)} \\
{}& = &{2\left( { – \frac{{\sqrt 2 }}{2} + i\frac{{\sqrt 2 }}{2}} \right)} \\
{}& = &{ – \sqrt 2 + \sqrt 2 i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\operatorname{cis} \frac{{9\pi }}{2}} \\
{}& = &{\cos \frac{{9\pi }}{2} + i\operatorname{sen} \frac{{9\pi }}{2}} \\
{}& = &{\cos \frac{\pi }{2} + i\operatorname{sen} \frac{\pi }{2}} \\
{}& = &i
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{9\operatorname{cis} 2\pi } \\
{}& = &{9\left( {\cos 2\pi + i\operatorname{sen} 2\pi } \right)} \\
{}& = &9
\end{array}$$
