Question

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

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

Answer 1

Answer: (C)

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

Answer 2

Writing in euler's form, we get
$e ^{ i \theta} e ^{2 i \theta} e ^{3 i \theta} \ldots e ^{ in \theta}=1$
$e ^{ i (\theta+2 \theta+3 \theta+\ldots \,n \theta}=1$
$e^{i \frac{n(n+1)}{2} \theta}=e^{2 m \pi}$
Therefore
$\frac{ n ( n +1)}{2} \theta=2 m \pi$
$\theta=\frac{4 m \pi}{n(n+1)}$