Question

the amplitude of 1-cos theta -i sin theta is

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{z=1-cos(\theta)-i\,sin(\theta)}$

$\tt{\implies\,|z|=\sqrt{\big(1-cos(\theta)\big)^2+sin^2(\theta)}}$

$\tt{\implies\,|z|=\sqrt{1+cos^2(\theta)-2\,cos(\theta)+sin^2(\theta)}}$

$\tt{\implies\,|z|=\sqrt{1+cos^2(\theta)+sin^2(\theta)-2\,cos(\theta)}}$

$\tt{\implies\,|z|=\sqrt{1+1-2\,cos(\theta)}}$

$\tt{\implies\,|z|=\sqrt{2-2\,cos(\theta)}}$

$\tt{\implies\,|z|=\sqrt{2\big(1-\,cos(\theta)\big)}}$

$\tt{\implies\,|z|=\sqrt{2\left(2\,sin^2\left(\dfrac{\theta}{2}\right)\right)}}$

$\tt{\implies\,|z|=\sqrt{4\,sin^2\left(\dfrac{\theta}{2}\right)\right)}}$

$\tt{\implies\,|z|=2\,sin\left(\dfrac{\theta}{2}\right)\right)}$