Escreva $z$ na forma trigonométrica
Números complexos: Infinito 12 A - Parte 3 Pág. 141 Ex. 47
Enunciado
Escreva $z$ na forma trigonométrica:
- $z = 1 – i\sqrt 3 $
- $z = – 1 + i$
- $z = – 5$
- $z = 3i$
- $z = \frac{1}{3} + \frac{1}{3}i$
- $z = – \sqrt 2 – \sqrt 6 i$
- $z = \frac{4}{{1 – i\sqrt 3 }}$
- $z = \frac{2}{{\sqrt 6 – i\sqrt 2 }}$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
z& = &{1 – i\sqrt 3 } \\
{}& = &{2\left( {\frac{1}{2} – \frac{{\sqrt 3 }}{2}i} \right)} \\
{}& = &{2\operatorname{cis} \left( { – \frac{\pi }{3}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{ – 1 + i} \\
{}& = &{\sqrt 2 \left( { – \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}i} \right)} \\
{}& = &{\sqrt 2 \operatorname{cis} \left( {\frac{{3\pi }}{4}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{ – 5} \\
{}& = &{5\operatorname{cis} \pi }
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{3i} \\
{}& = &{3\operatorname{cis} \frac{\pi }{2}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\frac{1}{3} + \frac{1}{3}i} \\
{}& = &{\frac{{\sqrt 2 }}{3}\left( {\frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}i} \right)} \\
{}& = &{\frac{{\sqrt 2 }}{3}\operatorname{cis} \frac{\pi }{4}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{ – \sqrt 2 – \sqrt 6 i} \\
{}& = &{2\sqrt 2 \left( { – \frac{1}{2} – \frac{{\sqrt 3 }}{2}i} \right)} \\
{}& = &{2\sqrt 2 \operatorname{cis} \left( {\frac{{4\pi }}{3}} \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\frac{4}{{1 – i\sqrt 3 }}} \\
{}& = &{\frac{{4\operatorname{cis} \left( 0 \right)}}{{2\left( {\frac{1}{2} – \frac{{\sqrt 3 }}{2}i} \right)}}} \\
{}& = &{\frac{{4\operatorname{cis} \left( 0 \right)}}{{2\operatorname{cis} \left( { – \frac{\pi }{3}} \right)}}} \\
{}& = &{2\operatorname{cis} \left( {0 + \frac{\pi }{3}} \right)} \\
{}& = &{2\operatorname{cis} \frac{\pi }{3}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\frac{2}{{\sqrt 6 – i\sqrt 2 }}} \\
{}& = &{\frac{{2\operatorname{cis} \left( 0 \right)}}{{2\sqrt 2 \left( {\frac{{\sqrt 3 }}{2} – \frac{1}{2}i} \right)}}} \\
{}& = &{\frac{{2\operatorname{cis} \left( 0 \right)}}{{2\sqrt 2 \operatorname{cis} \left( { – \frac{\pi }{6}} \right)}}} \\
{}& = &{\frac{2}{{2\sqrt 2 }}\operatorname{cis} \frac{\pi }{6}} \\
{}& = &{\frac{{\sqrt 2 }}{2}\operatorname{cis} \frac{\pi }{6}}
\end{array}$$
