If < <2 and z=1+ +i , then write the value of |z|.

Question

If $\pi<\theta<2\pi$ and $\text{z}=1+\cos\theta+\text{i}\sin\theta,$ then write the value of $|\text{z}|.$

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Answer

$\pi<\theta<2\pi$ $\frac{\pi}{2}<\frac{\theta}{2}<\pi$ (Dividing by 2) $\text{z}=1+\cos\theta+\text{i}\sin\theta$ $\Rightarrow|\text{z}|=\sqrt{(1+\cos\theta)^2+\sin ^2\theta}$ $\Rightarrow|\text{z}|=\sqrt{1+\cos^2\theta+2\cos\theta+\sin ^2\theta}$ $\Rightarrow|\text{z}|=\sqrt{1+1+2\cos\theta}$ $\Rightarrow|\text{z}|=\sqrt{2(1+2\cos\theta)}$ $\Rightarrow|\text{z}|=\sqrt{2\times2\cos^2\frac{\theta}{2}}$ $\Rightarrow|\text{z}|=2\sqrt{\cos^2\frac{\theta}{2}}$ $\Rightarrow|\text{z}|=-2\cos\frac{\theta}{2} \ \big[\text{Since}\frac{\pi}{2}<\frac{\theta}{2}<\pi,\cos\frac{\theta}{2} \ \text{is negative}\Big]$

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