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

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

Answer 1

The correct option is C

$\frac{4m\mathrm{\pi }}{n\left(n+1\right)}$

Explanation for the correct option:

Find the value of $\theta$:

Given, $(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )...(\cos n\theta +i\sin n\theta )=1$

$\Rightarrow$ $e^{i\theta }{e}^{2i\theta }{e}^{3i\theta }\dots .e^{in\theta }=1$ ; [Euler's form]

$\Rightarrow$ $e^{i(\theta +2\theta +3\theta +\dots n\theta )}=1$

$\Rightarrow$ $e^{in(n+1)\theta /2}=e^{i2m\pi }$

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

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

Hence, Option ‘C’ is Correct.