Question

O número complexo (1+i)90 é:

Answer 1

Fórmula de Moivre

$\huge\boxed{\boxed{\mathsf{{z}^{n}={\rho}^{n}[\cos(n\theta)+i\sin(n\theta)]}}}$

$\mathsf{\rho=\sqrt{{1}^{2}+{1}^{2}}=\sqrt{2}}$

$\mathsf{\tan(\theta)=\dfrac{1}{1}=1}\\\mathsf{\theta=\dfrac{\pi}{4}}$

$\mathsf{{(1+i)}^{90}={(\sqrt{2})}^{90}[\cos(90\dfrac{\pi}{4})+i\sin(90\dfrac{\pi}{4})]}}}$

$\huge\boxed{\boxed{\mathsf{{(1+i)}^{90}={2}^{45}[i]}}}$