Search any question & find its solution
Question:
Answered & Verified by Expert
If $Z_{r}=\sin \frac{2 \pi}{11}-i \cos \frac{2 \pi}{11},$ then $\sum_{r=0}^{10} Z_{r}$ is equal to
Options:
Solution:
1209 Upvotes
Verified Answer
The correct answer is:
0
We have, $Z,=\sin \frac{2 \pi r}{11}-i \cos \frac{2 \pi}{11}$
$$
=-i\left(\cos \frac{2 \pi r}{11}+i \sin \frac{2 \pi r}{11}\right)
$$
$\therefore$
$$
\begin{aligned}
&=-i e^{\frac{i 2 \pi}{11}} \\
\sum_{r=0}^{10} Z_{r} &=-i \sum_{r=0}^{10} e^{\frac{i 2 \pi}{11}}
\end{aligned}
$$
$$
=-i \times 0=0
$$
$$
=-i\left(\cos \frac{2 \pi r}{11}+i \sin \frac{2 \pi r}{11}\right)
$$
$\therefore$
$$
\begin{aligned}
&=-i e^{\frac{i 2 \pi}{11}} \\
\sum_{r=0}^{10} Z_{r} &=-i \sum_{r=0}^{10} e^{\frac{i 2 \pi}{11}}
\end{aligned}
$$
$$
=-i \times 0=0
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.