Q224 / 270 · NDA 2011 (Phase 1) · Mathematics

If $z=1+\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}$ then what is $|z|$ equal to?

  • A
    $2 \cos \frac{\pi}{5}$
  • B
    $2 \sin \frac{\pi}{5}$
  • Correct answer
    C
    $2 \cos \frac{\pi}{10}$
  • D
    $2 \sin \frac{\pi}{10}$

Solution

$\begin{aligned} z & =1+\cos \frac{\pi}{5}+i \sin \frac{\pi}{5} \\ & =2 \cos ^2 \frac{\pi}{10}+2 \hat{i} \sin \frac{\pi}{10} \cdot \cos \frac{\pi}{10} \\ z & =2 \cos \frac{\pi}{10}\left\{\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right\}\end{aligned}$ $\therefore \quad|z|=2 \cos \frac{\pi}{10}$.

Hence, the correct answer is option: 'C'.

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