Question

1) Determine o argumento do número complexo acima !​

Answer 1

Argumento de um número complexo

Chama-se argumento de um número complexo

z=a+bi, não nulo,ao ângulo $\mathsf{\theta}$ tal que

$\boxed{\mathsf{cos(\theta)=\dfrac{a}{\rho}~~e~~sen(\theta)=\dfrac{b}{\rho}~onde~\rho=|z|}}$

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$\mathtt{a)}~\mathsf{1+\sqrt{3}i}\\\mathsf{\rho=\sqrt{1^2+(\sqrt{3})^2}}\\\mathsf{\rho=\sqrt{1+3}=\sqrt{4}=2}\\\mathsf{cos(\theta)=\dfrac{1}{2}}\\\mathsf{sen(\theta)=\dfrac{\sqrt{3}}{2}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{\pi}{3}}}}}}\\\mathtt{b)}~\mathsf{-1+\sqrt{3}i}\\\mathsf{\rho=\sqrt{(-1)^2+(\sqrt{3})^2}}\\\mathsf{\rho=\sqrt{1+3}=\sqrt{4}=2}\\\mathsf{cos(\theta)=-\dfrac{1}{2}}\\\mathsf{sen(\theta)=\dfrac{\sqrt{3}}{2}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{2\pi}{3}}}}}}$

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