Calcule o valor de
Números complexos: Infinito 12 A - Parte 3 Pág. 143 Ex. 56
Calcule o valor de: $${{{\left( {\frac{{\cos \theta – i\operatorname{sen} \theta }}{{\operatorname{sen} \theta + i\cos \theta }}} \right)}^5}}$$
Ora,
$$\begin{array}{*{20}{l}}
{{{\left( {\frac{{\cos \theta – i\operatorname{sen} \theta }}{{\operatorname{sen} \theta + i\cos \theta }}} \right)}^5}}& = &{{{\left( {\frac{{\operatorname{cis} \left( { – \theta } \right)}}{{\cos \left( {\frac{\pi }{2} – \theta } \right) + i\operatorname{sen} \left( {\frac{\pi }{2} – \theta } \right)}}} \right)}^5}} \\
{}& = &{{{\left( {\frac{{\operatorname{cis} \left( { – \theta } \right)}}{{\operatorname{cis} \left( {\frac{\pi }{2} – \theta } \right)}}} \right)}^5}} \\
{}& = &{\frac{{\operatorname{cis} \left( { – 5\theta } \right)}}{{\operatorname{cis} \left( {\frac{{5\pi }}{2} – 5\theta } \right)}}} \\
{}& = &{\operatorname{cis} \left( { – 5\theta – \frac{{5\pi }}{2} + 5\theta } \right)} \\
{}& = &{\operatorname{cis} \left( { – \frac{{5\pi }}{2}} \right)} \\
{}& = &{\operatorname{cis} \left( { – \frac{\pi }{2}} \right)} \\
{}& = &{ – i}
\end{array}$$
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