Question

The continued product of the four values of $\left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)^{3/ 4 }$ is

• A. 1
• B. -1
• C. 2
• D. -2

Answer 1

Answer: (A)

$\left[\left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)^{3 }\right]^{1/ 4 } = \left(\cos \pi +i \sin \pi \right)^{1/ 4} $
$= \left[\cos \left( 2k\pi + \pi\right) +i \sin \left( 2k\pi + \pi\right) \right]^{1 /4} , k = 0, 1 , 2 ,3 $
$= \cos\left(2k + 1\right) \frac{\pi}{4} + i \sin\left(2k+1\right) \frac{\pi}{4} , k = 0,1,2,3 $
The continued product of the four values is
$\cos\left(\frac{\pi}{4}+\frac{3\pi}{4}+\frac{5\pi}{4}+\frac{7\pi}{4}\right) $
$+ i \sin\left(\frac{\pi}{4} + \frac{3\pi}{4}+\frac{5\pi}{4}+\frac{7\pi}{4}\right)$
$ = \cos 4\pi + i \sin 4\pi = 1 + i \cdot 0 = 1$