Question

The amplitude of $\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)$

• A. $\pi/5$
• B. $2\pi/5$
• C. $\pi/10$
• D. $\pi/15$

Answer 1

Answer: (C)

$\pi/10$

Answer 2

Sol.

$\sin\frac{\pi}{5} + i(1 - \cos\frac{\pi}{5}) = 2\sin\frac{\pi}{10}.\cos\frac{\pi}{10} + i2\sin^{2}\frac{\pi}{10}$ $$= 2\sin\frac{\pi}{10}\left( \cos\frac{\pi}{10} + i\sin\frac{\pi}{10} \right)$$

For amplitude, $\tan\theta = \frac{\sin\frac{\pi}{10}}{\cos\frac{\pi}{10}} = \tan\frac{\pi}{10} \Rightarrow \theta = \frac{\pi}{10}.$