Question

escreva a forma trigonometria o número complexo z=2+2raiz3i

Answer 1

$Z=2+2\sqrt3i\\a=2\\b=2\sqrt3$

Primeiro, extraindo o módulo de Z:

$|Z|=\sqrt{a^2+b^2}\\\\|Z|=\sqrt{2^2+(2\sqrt3)^2}\\\\|Z|=\sqr{4+12}\\\\\boxed{|Z|=4}$

Calculando o argumento:

$cos\mbox{ }\theta=\frac{a}{|Z|}\mbox{ }sen\mbox{ }\theta=\frac{b}{|Z|}\\\\cos\mbox{ }\theta=\frac{2}{4}\mbox{ }sen\mbox{ }\theta=\frac{2\sqrt3}{4}\\\\\underbrace{cos\mbox{ }\theta=\frac{1}{2}\mbox{ }sen\mbox{ }\theta=\frac{\sqrt3}{2}}_{\theta=60^o\iff\theta=\frac{\pi}{3}}$

O argumento vale θ = π/3.

Calculando a forma trigonométrica:

$Z=|Z|(cos\mbox{ }\theta+i\cdot sen\mbox{ }\theta)\\\\Z=4\cdot[\,cos\mbox{ }(\,\frac{\pi}{3})\,+i\cdot sen\mbox{ }(\,\frac{\pi}{3})\,]\,$