Question

(1/2 + i radical 3 /2)^2024+(1/2 - i radical 3 /2)^2024=?

Va rog ajutati-ma!!

Answer 1

$\left(\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)^{2024}+\left(\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right)^{2024}= \\ \\ = \left(\cos\dfrac{\pi}{3}+i\sin \dfrac{\pi}{3}\right)^{2024}+\left[-\left(\cos\dfrac{2\pi}{3}+i\sin \dfrac{2\pi}{3}\right)\right]^{2024}\\ \\ = \left(\cos\dfrac{2024\pi}{3}+i\sin \dfrac{2024\pi}{3}\right)+\left(\cos\dfrac{4048\pi}{3}+i\sin \dfrac{4048\pi}{3}\right)$

$=\cos\left(674\pi+\frac{2\pi}{3}\right)+i\sin\left(674\pi+\frac{2\pi}{3}\right) +\cos\left(1349\pi+\frac{\pi}{3}\right)+i\sin\left(1349\pi+\frac{\pi}{3}\right) \\ \\ =-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}-i\dfrac{\sqrt 3}{2}\\ \\ =\boxed{-1}$

Answer 2

Răspuns:

..........................................................................

Explicație pas cu pas: