Question

Geometricamente e determine o argumento de cada complexo abaixo. A)z=√5-2i B)z=2i

Answer 1

Forma trigonométrica de um número complexo

$\large\boxed{\boxed{\boxed{\boxed{\mathsf{z=\rho\left[cos(\theta)+i\,sen(\theta)\right]}}}}}$

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$\mathtt{A)}~\mathsf{z=\sqrt{5}-2i}\\\mathsf{\rho=\sqrt{(\sqrt{5})^2+(-2)^2}=\sqrt{5+4}=3}\\\mathsf{sen(\theta)=-\dfrac{2}{3}}\\\mathsf{cos(\theta)=\dfrac{\sqrt{5}}{3}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=138^{\circ}10'48''}}}}}\\\mathtt{B)}~\mathsf{z=2i\implies~z=0+2i}\\\mathsf{\rho=\sqrt{0^2+2^2}=\sqrt{4}=2}\\\mathsf{cos(\theta)=\dfrac{0}{2}=0}\\\mathsf{sen(\theta)=\dfrac{2}{2}=1}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{\pi}{2}}}}}}$

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