Question

O módulo e o argumento do complexo z = 1/2 + i/2 estão na alternativa:

a) p = √2/3 e θ = π
b) p = √2 e θ = 45°
c) p = √2/2 e θ = 120°
d) p = √2 e θ = π
e) p = √2/2 e θ = π/4

Answer 1

Olá


Alternativa correta, letra E)  p = √2/2 e θ = π/4



$\displaystyle\mathsf{z= \frac{1}{2} + \frac{i}{2} }$


Calculando o módulo ρ


$\displaystyle\mathsf{\rho= \sqrt{ \left(\frac{1}{2} \right)^2+\left( \frac{1}{2} \right)^2} }\\\\\\\\\mathsf{\rho= \sqrt{ \frac{1}{4} +\frac{1}{4}} }\\\\\\\\\mathsf{\rho= \sqrt{ \frac{1}{2} } }\\\\\\\\\mathsf{\rho= \frac{\sqrt{1}}{\sqrt{2}} }\\\\\\\\\mathsf{\rho= \frac{1}{\sqrt{2}} \cdot { \frac{\sqrt{2}}{\sqrt{2}} }}\\\\\\\\\boxed{\mathsf{\rho= \frac{\sqrt{2}}{2} }}}$




Calculando o argumento θ

forma do número complexo
z = a + bi


$\displaystyle \mathsf{sin\theta = \frac{b}{\rho}~=~ \frac{ \frac{1}{2} }{ \frac{ \sqrt{2} }{2} }~=~ \frac{ 1}{\sqrt{2}}~=~ \boxed{\frac{\sqrt{2}}{2}} }\\\\\\\mathsf{cos\theta= \frac{a}{\rho} ~=~\frac{ \frac{1}{2} }{ \frac{ \sqrt{2} }{2} }~=~ \frac{ 1}{\sqrt{2}}~=~ \boxed{\frac{\sqrt{2}}{2}} }}$



O seno e o cosseno do angulo que correspondem a $\dfrac{\sqrt{2}}{2}$ equivale a 45º, ou seja, está no primeiro quadrante.


45º no ciclo trigonométrico equivale à π/4


Por tanto, a alternativa correta é letra E)