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
$\cos \left(\frac{\pi }{4}+\frac{3\pi }{4}+\frac{5\pi }{4}+\frac{7\pi }{4}\right)$
$+i\sin \left(\frac{\pi }{4}+\frac{3\pi }{4}+\frac{5\pi }{4}+\frac{7\pi }{4}\right)$
$=\cos 4\pi +i\sin 4\pi =1+i\cdot 0=1$