Represente na forma trigonométrica
Números complexos: Infinito 12 A - Parte 3 Pág. 96 Ex. 57
Enunciado
Represente na forma trigonométrica:
- $z = – 3\operatorname{cis} \theta $
- $z = 2\cos \theta – 2i\operatorname{sen} \theta $
- $z = – \cos \theta – i\operatorname{sen} \theta $
- $z = \frac{1}{{2\operatorname{cis} \left( {\frac{\pi }{3} – \theta } \right)}}$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
z& = &{ – 3\operatorname{cis} \theta } \\
{}& = &{3\operatorname{cis} \left( {\pi + \theta } \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{2\cos \theta – 2i\operatorname{sen} \theta } \\
{}& = &{2\cos \left( { – \theta } \right) + 2i\operatorname{sen} \left( { – \theta } \right)} \\
{}& = &{2\operatorname{cis} \left( { – \theta } \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{ – \cos \theta – i\operatorname{sen} \theta } \\
{}& = &{\cos \left( {\pi + \theta } \right) + i\operatorname{sen} \left( {\pi + \theta } \right)} \\
{}& = &{\operatorname{cis} \left( {\pi + \theta } \right)}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
z& = &{\frac{1}{{2\operatorname{cis} \left( {\frac{\pi }{3} – \theta } \right)}}} \\
{}& = &{\frac{1}{2}\operatorname{cis} \left( { – \frac{\pi }{3} + \theta } \right)} \\
{}& = &{\frac{1}{2}\operatorname{cis} \left( {\theta – \frac{\pi }{3}} \right)}
\end{array}$$
