Question

Escreva na forma trigonometrica o numero complexo z = 1+ Raiz de 3i

Answer 1

Z=1+√3
Z=a+bi

a=1
b=√3

Resoluçao:

|Z|²=a²+b²

|Z|²=1²+(√3)
|Z|²=1+3
|Z|²=4
|Z|=√4
|Z|=2

achando o angulo:

tg=b/a
tg=√3/1
tg=60°


Z=|Z|(cosx+isenx)
Z=2(cos60+isen60)

ou

Z=2(cosπ/3+isenπ/3)

Answer 2

Resposta:

d) $2\left(cos\frac{\pi }{3}+isen\frac{\pi }{3}\right)$

Alternativas:

a) $2\left(cos\frac{2\pi }{3}+isen\frac{2\pi }{3}\right)$  

b) $3\left(cos\frac{\pi }{3}+isen\frac{\pi }{3}\right)$  

c) $2\left(cos\frac{4\pi }{3}+isen\frac{4\pi }{3}\right)$

d) $2\left(cos\frac{\pi }{3}+isen\frac{\pi }{3}\right)$

e) $3\left(cos\frac{\pi }{3}+isen\frac{\pi }{3}\right)$

Explicação passo-a-passo:

Sendo $z=1+\sqrt{3} i$, temos:

$\left|z\right|=\sqrt{a^2+b^2}=\sqrt{\left(1\right)^2+\left(\sqrt{3}\right)^2}=2$

$\left \{ {{cos\theta =\frac{a}{\left|z\right|}=\frac{1}{2}} \atop {sen\theta =\frac{b}{\left|z\right|}=\frac{\sqrt{3}}{2}}} \right. \rightarrow \theta =\frac{\pi }{3}$  

Portanto, $z=2\left(cos\frac{\pi }{3}+isen\frac{\pi }{3}\right)$.