Exercicios de Matematica 12 ANO - Números Complexos - Exercício 4

Considere os números e $w = 2{\rm{cis}}\left( {\frac{\pi }{3}} \right)$ .

4.1. Represente na forma algébrica o complexo $t = \frac{r}{w}$ .

4.2. Resolva, em $\mathbb{C}$ , a equação $i{z^4} = w$ .

4.3. Sabe-se que é uma das raízes cúbicas de um complexo , determine as outras raízes cúbicas de .

Resolução do exercício de Matemática:

4.1. $w = 2\operatorname{cis} \left( {\frac{\pi }{3}} \right) = 2\left( {\cos \left( {\frac{\pi }{3}} \right) + i\sin \left( {\frac{\pi }{3}} \right)} \right) = 2\left( {\frac{1}{2} + \frac{{\sqrt 3 }}{2}i} \right) = 1 + \sqrt 3 i$

$t = \frac{r}{w} = \frac{{4 + 2i}}{{1 + \sqrt 3 i}} = \frac{{\left( {4 + 2i} \right)\left( {1 - \sqrt 3 i} \right)}}{{\left( {1 + \sqrt 3 i} \right)\left( {1 - \sqrt 3 i} \right)}} = \frac{{4 - 4\sqrt 3 i + 2i - 2\sqrt 3 {i^2}}}{{1 - 3{i^2}}} =$

$= \frac{{4 - 4\sqrt 3 i + 2i - 2\sqrt 3 \left( { - 1} \right)}}{{1 - 3\left( { - 1} \right)}} = \frac{{4 + 2\sqrt 3 + 2i - 4\sqrt 3 i}}{4} =$

$= \frac{{2 + \sqrt 3 }}{2} + \frac{{1 - 2\sqrt 3 }}{2}i$

4.2. $i{z^4} = w \Leftrightarrow {z^4} = \frac{w}{i} \Leftrightarrow {z^4} = \frac{{2\operatorname{cis} \left( {\frac{\pi }{3}} \right)}}{{1\operatorname{cis} \left( {\frac{\pi }{2}} \right)}} \Leftrightarrow {z^4} = \frac{2}{1}\operatorname{cis} \left( {\frac{\pi }{3} - \frac{\pi }{2}} \right) \Leftrightarrow$

$\Leftrightarrow {z^4} = 2\operatorname{cis} \left( { - \frac{\pi }{6}} \right) \Leftrightarrow z = \sqrt[4]{2}\operatorname{cis} \left( {\frac{{ - \frac{\pi }{6} + 2k\pi }}{4}} \right),k \in \left\{ {0,1,2,3} \right\} \Leftrightarrow$

$\Leftrightarrow z = \sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}} + \frac{{k\pi }}{2}} \right),k \in \left\{ {0,1,2,3} \right\}$

$k = 0,{\text{ }}{z_1} = \sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}}} \right)$

$k = 1,{\text{ }}{z_2} = \sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}} + \frac{\pi }{2}} \right) = \sqrt[4]{2}\operatorname{cis} \left( {\frac{{11\pi }}{{24}}} \right)$

$k = 2,{\text{ }}{z_2} = \sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}} + \frac{{2\pi }}{2}} \right) = \sqrt[4]{2}\operatorname{cis} \left( {\frac{{23\pi }}{{24}}} \right)$

$k = 3,{\text{ }}{z_2} = \sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}} + \frac{{3\pi }}{2}} \right) = \sqrt[4]{2}\operatorname{cis} \left( {\frac{{35\pi }}{{24}}} \right)$

$S = \left\{ {\sqrt[4]{2}\operatorname{cis} \left( { - \frac{\pi }{{24}}} \right),\sqrt[4]{2}\operatorname{cis} \left( {\frac{{11\pi }}{{24}}} \right),\sqrt[4]{2}\operatorname{cis} \left( {\frac{{23\pi }}{{24}}} \right),\sqrt[4]{2}\operatorname{cis} \left( {\frac{{35\pi }}{{24}}} \right)} \right\}$