Question

50 PONTOS!

1) Escreva na forma trigonometrica os seguintes numeros complexos

a) $7i$

b)$1 - \sqrt{3i}$


2) Expresse na forma algebrica:

3 · $( cos \frac{\pi }{6} + i VEZES sen \frac{\pi }{6} )$

Answer 1

$\Large\boxed{\begin{array}{l}\rm Z=a+bi\\\underline{\sf M\acute odulo~de~um~n\acute umero~complexo}\\\huge\boxed{\boxed{\boxed{\boxed{\rm\rho=\sqrt{a^2+b^2} }}}}\\\underline{\sf Argumento~de~um~n\acute umero~complexo}\\\rm \acute E~o~\hat angulo~\theta~tal~que\\\rm sen(\theta)=\dfrac{a}{\rho}~e~cos(\theta)=\dfrac{b}{\rho}\\\underline{\sf Forma~trigonom\acute etrica~de~um~n\acute umero~complexo}\\\huge\boxed{\boxed{\boxed{\boxed{\rm Z=\rho[cos(\theta)+i~sen(\theta)]}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\sf 1)\\\sf a)\\\rm \rho=\sqrt{7^2}=7\\\begin{cases}\rm cos(\theta)=\dfrac{0}{7}=0\\\\\rm sen(\theta)=\dfrac{7}{7}=1\end{cases}\longrightarrow \rm \theta=2\pi\\\rm z=7[cos(2\pi)+i~sen(2\pi)]\end{array}}$

$\large\boxed{\begin{array}{l}\sf b)\\\rm z=1-\sqrt{3}i\\\rm \rho=\sqrt{1^2+(\sqrt{3})^2}=\sqrt{1+3}=\sqrt{4}=2\\\begin{cases}\rm cos(\theta)=\dfrac{1}{2}\\\\\rm sen(\theta)=-\dfrac{\sqrt{3}}{2}\end{cases}\longrightarrow\rm\theta=\dfrac{5\pi}{3}\\\\\rm z=2\bigg[cos\bigg(\dfrac{5\pi}{3}\bigg)+i~sen\bigg(\dfrac{5\pi}{3}\bigg)\bigg]\end{array}}$

$\large\boxed{\begin{array}{l}\sf 2)\\\rm z=3\bigg[cos\bigg(\dfrac{\pi}{6}\bigg)+i~sen\bigg(\dfrac{\pi}{6}\bigg)\bigg]\\\\\rm z=3\bigg[\dfrac{\sqrt{3}}{2}+i\cdot\dfrac{1}{2}\bigg]\\\\\rm z=\dfrac{3\sqrt{3}}{2}+\dfrac{3}{2}i\end{array}}$