Considere os seguintes números complexos
Números complexos: Infinito 12 A - Parte 3 Pág. 142 Ex. 53
Considere $$\begin{array}{*{20}{c}}
{{z_1} = \frac{{\sqrt 3 }}{2} – \frac{1}{2}i}&{\text{e}}&{{z_2} = \operatorname{cis} \frac{{3\pi }}{4}}
\end{array}$$
- Determine ${z_1}.{z_2}$, na forma trigonométrica e na forma algébrica.
- Utilizando os resultados obtidos na alínea anterior, deduza os valores exatos de $\cos \frac{{7\pi }}{{12}}$ e $\operatorname{sen} \frac{{7\pi }}{{12}}$.
- Obtenha os valores de $\cos \frac{{7\pi }}{{12}}$ e $\operatorname{sen} \frac{{7\pi }}{{12}}$ utilizando outro processo.
(Sugestão: $\frac{{7\pi }}{{12}} = \frac{\pi }{4} + \frac{\pi }{3}$)
$$\begin{array}{*{20}{c}}
{{z_1} = \frac{{\sqrt 3 }}{2} – \frac{1}{2}i}&{\text{e}}&{{z_2} = \operatorname{cis} \frac{{3\pi }}{4}}
\end{array}$$
- Ora,
$$\begin{array}{*{20}{l}}
{{z_1}.{z_2}}& = &{\left( {\frac{{\sqrt 3 }}{2} – \frac{1}{2}i} \right) \times \operatorname{cis} \frac{{3\pi }}{4}} \\
{}& = &{\operatorname{cis} \left( { – \frac{\pi }{6}} \right) \times \operatorname{cis} \frac{{3\pi }}{4}} \\
{}& = &{\operatorname{cis} \left( { – \frac{\pi }{6} + \frac{{3\pi }}{4}} \right)} \\
{}& = &{\operatorname{cis} \left( {\frac{{7\pi }}{{12}}} \right)}
\end{array}$$$$\begin{array}{*{20}{l}}
{{z_1}.{z_2}}& = &{\left( {\frac{{\sqrt 3 }}{2} – \frac{1}{2}i} \right) \times \operatorname{cis} \frac{{3\pi }}{4}} \\
{}& = &{\left( {\frac{{\sqrt 3 }}{2} – \frac{1}{2}i} \right) \times \left( { – \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}i} \right)} \\
{}& = &{ – \frac{{\sqrt 6 }}{4} + \frac{{\sqrt 6 }}{4}i + \frac{{\sqrt 2 }}{4}i + \frac{{\sqrt 2 }}{4}} \\
{}& = &{\frac{{ – \sqrt 6 + \sqrt 2 }}{4} + \frac{{\sqrt 6 + \sqrt 2 }}{4}i}
\end{array}$$ - Como $${z_1}.{z_2} = \operatorname{cis} \left( {\frac{{7\pi }}{{12}}} \right) = \cos \frac{{7\pi }}{{12}} + i\operatorname{sen} \frac{{7\pi }}{{12}}$$ então $$\cos \frac{{7\pi }}{{12}} = \frac{{ – \sqrt 6 + \sqrt 2 }}{4}$$ e $$\operatorname{sen} \frac{{7\pi }}{{12}} = \frac{{\sqrt 6 + \sqrt 2 }}{4}$$
- Ora,
$$\begin{array}{*{20}{l}}
{\cos \frac{{7\pi }}{{12}}}& = &{\cos \left( {\frac{\pi }{4} + \frac{\pi }{3}} \right)} \\
{}& = &{\cos \frac{\pi }{4}\cos \frac{\pi }{3} – \operatorname{sen} \frac{\pi }{4}\operatorname{sen} \frac{\pi }{3}} \\
{}& = &{\frac{{\sqrt 2 }}{2} \times \frac{1}{2} – \frac{{\sqrt 2 }}{2} \times \frac{{\sqrt 3 }}{2}} \\
{}& = &{\frac{{ – \sqrt 6 + \sqrt 2 }}{4}}
\end{array}$$$$\begin{array}{*{20}{l}}
{\operatorname{sen} \frac{{7\pi }}{{12}}}& = &{\operatorname{sen} \left( {\frac{\pi }{4} + \frac{\pi }{3}} \right)} \\
{}& = &{\operatorname{sen} \frac{\pi }{4}\cos \frac{\pi }{3} + \operatorname{sen} \frac{\pi }{3}\cos \frac{\pi }{4}} \\
{}& = &{\frac{{\sqrt 2 }}{2} \times \frac{1}{2} + \frac{{\sqrt 3 }}{2} \times \frac{{\sqrt 2 }}{2}} \\
{}& = &{\frac{{\sqrt 6 + \sqrt 2 }}{4}}
\end{array}$$
