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Definite Integral as Limit of Sum

PASSAGE-3:Let...

Question

PASSAGE-3:
Let $A_{1}, A_{2}, A_{3} \dots A_{n}$ be a regular polygon of $n$ sides whose center is origin $O$ . Let the complex numbers representing vertices $A_{1}, A_{2}, A_{3} \dots A_{n}$ be $z_{1}, z_{2}, z_{3} \dots z_{n}$ respectively. Let $O A_{1} = O A_{2} = \dots = O A_{n} = 1$ .

The distances $A_{1} A_{j} (j = 2, 3, \dots, n)$ must be

sin (\frac{J π}{n})

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2 sin (\frac{(j - 1) π}{n})

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cos (\frac{J π}{n})

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2 sin (\frac{(j + 1) π}{n})

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Solution

The correct option is B $2 sin (\frac{(j - 1) π}{n})$
By sine rule in $Δ A_{1} O A_{j}$
$\frac{a}{sin θ_{1}} = \frac{1}{sin (\frac{π - θ_{1}}{2})}$
$\Rightarrow \frac{a}{sin \frac{(j - 1) 2 π}{n}}$
$= \frac{1}{sin (\frac{π}{2} - \frac{(j - 1) π}{n})}$
$\Rightarrow \frac{a}{2 sin \frac{(j - 1) π}{n} cos \frac{(j - 1) π}{n}} = \frac{1}{cos \frac{(j - 1) π}{n}}$
$\Rightarrow a = 2 sin \frac{(j - 1) π}{n}$ $\to$ Side

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