The amplitude of the complex number z = sin a | vedclass

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(B) Given $z = \sin \alpha   i(1 - \cos \alpha )$.The amplitude (argument) of $z = x   iy$ is given by $	heta = 	an^{-1}\left(\frac{y}{x}ight)$.He

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$\frac{\alpha }{2}$

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(B) Given $z = \sin \alpha   i(1 - \cos \alpha )$.The amplitude (argument) of $z = x   iy$ is given by $	heta = 	an^{-1}\left(\frac{y}{x}ight)$.Here,$x = \sin \alpha$ and $y = 1 - \cos \alpha$.Therefore,$	ext{amp}(z) = 	an^{-1}\left(\frac{1 - \cos \alpha}{\sin \alpha}ight)$.Using trigonometric identities $1 - \cos \alpha = 2\sin^2\left(\frac{\alpha}{2}ight)$ and $\sin \alpha = 2\sin\left(\frac{\alpha}{2}ight)\cos\left(\frac{\alpha}{2}ight)$:$	ext{amp}(z) = 	an^{-1}\left(\frac{2\sin^2\left(\frac{\alpha}{2}ight)}{2\sin\left(\frac{\alpha}{2}ight)\cos\left(\frac{\alpha}{2}ight)}ight) = 	an^{-1}\left(	an\left(\frac{\alpha}{2}ight)ight) = \frac{\alpha}{2}$.

The amplitude of the complex number $z = \sin \alpha + i(1 - \cos \alpha )$ is

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