Question

Escreva o número complexo z = 1 + i na forma trigonométrica.

Answer 1

Seja Z = a+b.i

Na forma trigonométrica :

$\text Z = |\text Z|.[\te{Cos}(\theta) + \text{i.Sen}(\theta)]$

sendo :

$|\text Z| = \sqrt{ \text a^2 + \text b^2}$

$\displaystyle \text{Cos}(\theta) = \frac{\text a }{|\text Z|}$

$\displaystyle \text{Sen}(\theta) = \frac{\text b }{|\text Z|}$

Temos Z = 1 + i e queremos na forma trigonométrica, então :

Z = 1 + 1.i , então : a = 1 , b = 1

$|\text Z| = \sqrt{1^2+1^2 } = \sqrt{2}$

$\displaystyle \text{Cos}(\theta) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

$\displaystyle \text{Sen}(\theta) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Então $\displaystyle \theta= \frac{\pi}{4}$

Escrevendo na forma trigonométrica  :

$\huge\boxed{\text Z = \sqrt{2}.[\text{Cos}(\frac{\pi}{4})+\text{i.Sen}(\frac{\pi}{4})] }\checkmark$