If z=1- +i , then |z|= - Vidyadip

MCQ

If $\text{z}=1-\cos\theta+\text{i}\sin\theta,$ then $|\text{z}|=$

A
$2\sin\frac{\theta}{2}$
B
$2\cos\frac{\theta}{2}$
✓
$2\Big|\sin\frac{\theta}{2}\Big|$
D
$2\Big|\cos\frac{\theta}{2}\Big|$

✓

Answer

Correct option: C.

$2\Big|\sin\frac{\theta}{2}\Big|$

$\therefore\text{z}=1-\cos\theta+\text{i}\sin\theta$
$\Rightarrow|\text{z}|=\sqrt{(1-\cos\theta)^2+\sin^2\theta}$
$\Rightarrow|\text{z}|=\sqrt{1+\cos^2\theta-2\cos\theta+\sin^2\theta}$
$\Rightarrow|\text{z}|=\sqrt{1+1-2\cos\theta}$
$\Rightarrow|\text{z}|=\sqrt{2(1-2\cos\theta)}$
$\Rightarrow|\text{z}|=\sqrt{4\sin^2\frac{\theta}{2}}$
$\Rightarrow|\text{z}|=2\Big|\sin\frac{\theta}{2}\Big|$

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