Question

No plano complexo (Plano de Argand-Gauss), o afixo de um número complexo z é dado por um ponto P(-2,2). Calcule o módulo e o argumento desse número complexo z.
Sua resposta

Answer 1

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

$\Large\boxed{\begin{array}{l}\sf P(-2,2)\implies Z=-2+2i\\\sf \rho=\sqrt{(-2)^2+2^2}\\\sf\rho=\sqrt{4+4}\\\sf \rho=\sqrt{2\cdot4}\\\huge\boxed{\boxed{\boxed{\boxed{\sf\rho=2\sqrt{2}}}}}\\\sf cos(\theta)=\dfrac{-\backslash\!\!\!2}{\backslash\!\!\!2\sqrt{2}}=-\dfrac{\sqrt{2}}{2}\\\\\sf sen(\theta)=\dfrac{\backslash\!\!\!2}{\backslash\!\!\!2\sqrt{2}}=\dfrac{\sqrt{2}}{2}\\\huge\boxed{\boxed{\boxed{\boxed{\sf \theta=\dfrac{3\pi}{4}}}}}\end{array}}$