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

COMEDKCOMEDK 2007Complex Numbers and Quadratic Equations

Solution:

Let $Z = \sin \frac{\pi}{5} + i\left( 1 - \cos \frac{\pi}{5}\right) $
Putting $\sin \frac{\pi}{5} =r \cos \theta $ ...(i)
and $1 -\cos \frac{\pi}{5} = r \sin \theta$...(ii)
$ \therefore \:\:\:\: \tan \theta = \frac{1-\cos \frac{\pi}{5}}{\sin \frac{\pi}{5}} = \frac{ 2 \sin^{2} \frac{\pi}{10}}{2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}}$
$ \Rightarrow \tan \theta = \tan \frac{\pi}{10} \Rightarrow \theta = \frac{\pi}{10} $