Question

If $(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta ).....(\cos n\theta +i\sin n\theta )=1$, then the value of $\theta$ is, $mϵN$

• A. $4m\pi$
• B. $\frac{2m\pi }{n(n+1)}$
• C. $\frac{4m\pi }{n(n+1)}$
• D. $\frac{m\pi }{n(n+1)}$

Answer 1

The correct option is C $\frac{4m\pi }{n(n+1)}$

Writing in euler's form, we get

$e^{i\theta }{e}^{2i\theta }{e}^{3i\theta }...e^{in\theta }=1$

$e^{i(\theta +2\theta +3\theta +...n\theta }=1$

$e^{i\frac{n(n+1)}{2}\theta }=e^{2m\pi }$

Therefore

$\frac{n(n+1)}{2}\theta =2m\pi$

$\theta =\frac{4m\pi }{n(n+1)}$