Question

Sem recorrer à calculadora, determine o valor da expressão abaixo.
$\dfrac{\left(1+\sqrt{3}\,i\right)^2+\left(2\;cis\!\left(\dfrac{\pi}{12}\right)\right)^4}{6\,cis(\pi)}$

Nota: A função cis(x) é dada por $cis(x)=\cos(x)+i\sin(x)$.

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\sf \dfrac{(1 + \sqrt{3}\:i)^2 + \left(2\:cis\:\left(\dfrac{\pi }{12}\right)\right)^4}{6\:cis\:(\pi )}$

$\sf \dfrac{(1 + \sqrt{3}\:i)^2 + \left(2\:.\:\left(cos\:\left(\dfrac{\pi }{12}\right) + i\:.\:sen\:\left(\dfrac{\pi }{12}\right)\right)\right)^4}{6\:.\:\left(cos\:(\pi ) + i\:.\:sen\:(\pi )\right)}$

$\sf \dfrac{(1 + 2\sqrt{3}\:i + (\sqrt{3}\:i)^2) + \left(2^4\:.\:\left(cos\:\left(4\:.\:\dfrac{\pi }{12}\right) + i\:.\:sen\:\left(4\:.\:\dfrac{\pi }{12}\right)\right)\right)}{6\:.\:\left(cos\:(\pi ) + i\:.\:sen\:(\pi )\right)}$

$\sf \dfrac{(-2 + 2\sqrt{3}\:i) + \left(16\:.\:\left(cos\:\left(\dfrac{\pi }{3}\right) + i\:.\:sen\:\left(\dfrac{\pi }{3}\right)\right)\right)}{6\:.\:\left(cos\:(\pi ) + i\:.\:sen\:(\pi )\right)}$

$\sf \dfrac{(-2 + 2\sqrt{3}\:i) + \left(16\:.\:\left(\dfrac{1}{2} + i\:.\:\dfrac{\sqrt{3}}{2}\right)\right)}{6\:.\:\left(-1 + i\:.\:0)}$

$\sf \dfrac{(-2 + 2\sqrt{3}\:i) + (8 + 8\sqrt{3}\:i)}{-6}$

$\sf \dfrac{6 + 10\sqrt{3}\:i}{-6}$

$\boxed{\boxed{\sf -1 - \dfrac{5\sqrt{3}\:i}{3}}}$