Question

20. Escreva a forma trigonométrica do número
complexo

Answer 1

Resposta:

Explicação passo-a-passo:

$\displaystyle \frac{1+i}{i} = \frac{(1+i)}{i} .\frac{i}{i} =\frac{i+i^2}{i^2} =\frac{i-1}{-1} =1-i$

Answer 2

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módulo de um número complexo

$\Large\boxed{\begin{array}{l}\sf Seja~z=a+bi~um~n\acute umero~complexo.\\\sf o~m\acute odulo~de~z~\acute e~dado~por:\\\Huge\boxed{\boxed{\boxed{\boxed{\sf \rho=\sqrt{a^2+b^2}}}}}\end{array}}$

Argumento de um número complexo

$\Large\boxed{\begin{array}{l}\sf \acute e~o~\hat angulo~\theta~tal~que\\\boxed{\boxed{\sf cos(\theta)=\dfrac{a}{\rho}}}~e~\boxed{\boxed{\sf sen(\theta)=\dfrac{b}{\rho}}}\end{array}}$

Forma trigonométrica de  um número complexo

$\Large\boxed{\begin{array}{l}\sf \acute e~a~representac_{\!\!,}\tilde ao~do~complexo~z=a+bi~\\\sf em~func_{\!\!,}\tilde ao~de~seu~m\acute odulo~e~de~seu~argumento.\\\sf a~forma~trigonom\acute etrica~de~z~\acute e:\\\huge\boxed{\boxed{\boxed{\boxed{\sf z=\rho[cos(\theta)+i~sen(\theta)]}}}}\end{array}}$

$\sf z=\dfrac{1+i}{i}\\\sf z=\dfrac{(1+i)}{i}\cdot\dfrac{i}{i}\\\sf z=\dfrac{i+i^2}{i^2}\\\sf z=\dfrac{i-1}{-1}\implies z= 1-i\\\sf \rho=\sqrt{1^2+(-1)^2}\\\sf\rho=\sqrt{1+1}\\\sf\rho=\sqrt{2}$

$\begin{cases}\sf sen(\theta)=\dfrac{-1}{\sqrt{2}}=-\dfrac{\sqrt{2}}{2}\\\sf cos(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\end{cases}\implies \theta=\dfrac{7\pi}{4}\\\huge\boxed{\boxed{\boxed{\boxed{\sf z=\sqrt{2}\bigg[cos\bigg(\dfrac{7\pi}{4}\bigg)+i~sen\bigg(\dfrac{7\pi}{4}\bigg)\bigg]}}}}$