Question

The value of $2{\Pi }_{p=1}^6(\sin \frac{2p\pi }{7}-i\cos \frac{2p\pi }{7})$ is

• A. $2i$
• B. $-2i$
• C. $2$
• D. $1$

Answer 1

The correct option is B $-2i$

$y=2{\Pi }_{p=1}^6\left(\sin \frac{2p\pi }{7}-i\cos \frac{2p\pi }{7}\right)iy=2{\Pi }_{p=1}^6\left(i\sin \frac{2p\pi }{7}-i^2\cos \frac{2p\pi }{7}\right)iy=2{\Pi }_{p=1}^6(\cos \frac{2p\pi }{7}+i\sin \frac{2p\pi }{7})iy=2e^{{\sum }_{p=1}^{6}i\left(\frac{2p\pi }{7}\right)}iy={2e}^{\frac{2\pi i}{7}{\sum }_{p=1}^{6}p}iy={2e}^{\frac{2\pi i}{7}\times 21}=e^{i6\pi }iy=2(\cos 6\pi +i\sin 6\pi )iy=2i^{2}y=2iy=-2i$

So, option B is correct.