Question

$\mathrm{If}z=1-\mathrm{cos\theta }+i\mathrm{sin\theta },\mathrm{then}\left|z\right|=$
(a) $2\sin \frac{\mathrm{\theta }}{2}$
(b) $2\cos \frac{\mathrm{\theta }}{2}$
(c) $2\left|\sin \frac{\mathrm{\theta }}{2}\right|$
(d) $2\left|\cos \frac{\mathrm{\theta }}{2}\right|$

Answer 1

(c) $2\left|\sin \frac{\theta }{2}\right|$

$$\begin{gathered} \because z=1-\cos \theta +i\sin \theta \\ \Rightarrow \left|z\right|=\sqrt{{\left(1-\cos \theta \right)}^2+{\sin }^2\theta } \\ \Rightarrow \left|z\right|=\sqrt{1+{\cos }^2\theta -2\cos \theta +{\sin }^2\theta } \\ \Rightarrow \left|z\right|=\sqrt{1+1-2\cos \theta } \\ \Rightarrow \left|z\right|=\sqrt{2\left(1-\cos \theta \right)} \\ \Rightarrow \left|z\right|=\sqrt{4{\sin }^2\frac{\theta }{2}} \\ \Rightarrow \left|z\right|=2\left|\sin {\displaystyle \frac{\theta }{2}}\right| \end{gathered}$$