Question

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

• A. $\frac{\pi }{5}$
• B. $\frac{2\pi }{5}$
• C. $\frac{\pi }{10}$
• D. $\frac{\pi }{15}$

Answer 1

The correct option is C $\frac{\pi }{10}$

Given,

$\Rightarrow \sin \frac{\pi }{5}+i(1-\cos \frac{\pi }{5})$
$=2\sin \frac{\pi }{10}\cos \frac{\pi }{10}+i2{\sin }^2\frac{\pi }{10}$ $[\because \sin 2\theta =2\sin \theta \cos \theta 2{\sin }^2\theta =1-\cos \theta ]$

$=2\sin \frac{\pi }{10}(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10})$

Then

$Amplitude=\tan \theta =\frac{y}{x}$

$\tan \theta =\frac{\sin \frac{\pi }{10}}{\cos \frac{\pi }{10}}=\tan \frac{\pi }{10}$

$\Rightarrow \theta =\frac{\pi }{10}$