Question

If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of $\left|z\right|$.

Answer 1

$$\begin{gathered} \mathrm{\pi }<\mathrm{\theta }<2\mathrm{\pi } \\ \frac{\mathrm{\pi }}{2}<\frac{\theta }{2}<\mathrm{\pi }\left(\mathrm{Dividing}\mathrm{by}2\right) \\ z=1+\cos \theta +\mathrm{isin}\theta \\ \Rightarrow \left|z\right|=\sqrt{{\left(1+\cos \theta \right)}^2+{\sin }^2\theta } \\ \Rightarrow \left|z\right|=\sqrt{1+{\cos }^2\theta +2\cos \theta +{\sin }^2\theta } \\ \Rightarrow \left|z\right|=\sqrt{1+1+2\cos \theta } \\ \Rightarrow \left|z\right|=\sqrt{2\left(1+\cos \theta \right)} \\ \Rightarrow \left|z\right|=\sqrt{2\times 2{\cos }^2\frac{\theta }{2}} \\ \Rightarrow \left|z\right|=2\sqrt{{\cos }^2\frac{\theta }{2}} \\ \Rightarrow \left|z\right|=-2\cos \frac{\theta }{2}\left[\mathrm{Since}\frac{\mathrm{\pi }}{2}<\frac{\theta }{2}<\mathrm{\pi },\cos \frac{\theta }{2}\mathrm{is}\mathrm{negative}\right] \end{gathered}$$