Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

3. $2\Big|\sin\frac{\theta}{2}\Big|$

Solution:

$\therefore\text{z}=1-\cos\theta+\text{i}\sin\theta$

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

$\Rightarrow|\text{z}|=\sqrt{1+\cos^2\theta-2\cos\theta+\sin^2\theta}$

$\Rightarrow|\text{z}|=\sqrt{1+1-2\cos\theta}$

$\Rightarrow|\text{z}|=\sqrt{2(1-2\cos\theta)}$

$\Rightarrow|\text{z}|=\sqrt{4\sin^2\frac{\theta}{2}}$

$\Rightarrow|\text{z}|=2\Big|\sin\frac{\theta}{2}\Big|$