Question

The general value of the real angle \(\theta\), which satisfies the equation, \((\cos \theta+i \sin \theta)(\cos 2 \theta+i \sin 2 \theta) \ldots \ldots .\) \((\cos n \theta+i \sin n \theta)=1\) is given by, (assuming \(k\) is an integer)
(A) \(\frac{2 k \pi}{n+2}\)
(B) \(\frac{4 k \pi}{n(n+1)}\)
(C) \(\frac{4 k \pi}{n+1}\)
(D) \(\frac{6 k \pi}{n(n+1)}\)

Answer 1

Ans: (B)
Hint \(: \mathrm{e}^{\mathrm{i} \theta} \cdot \mathrm{e}^{\mathrm{i}(2 \theta)} \cdot \mathrm{e}^{\mathrm{i}(3 \theta)} \ldots \mathrm{e}^{\mathrm{i}(\mathrm{n} \theta)}=1 \Rightarrow \mathrm{e}^{i\left(\frac{(\mathrm{n}(\mathrm{n}+1))}{2}\right)}=\mathrm{e}^{\mathrm{i} 2 \mathrm{k} \pi} \Rightarrow \theta=\frac{4 \mathrm{k} \pi}{\mathrm{n}(\mathrm{n}+1)}\)

Answer 2

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Answer 3

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