If w -cos-n- i sin- n - then value of 1- w - w 2- w n -1 isA- 0B- 1- i -2 -C- 1- i tan-2 nD- 1- i cos-2 -

The correct option is B 1 + iπ/2n To find the value of 1 + w + w^2 + ⋯ + w^n 1 This is a Geometric Progression with first term as 1 and and common ratio as w ⇒ ...

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Standard XII

Mathematics

Circular Measurement of Angle

If w =cosπ/π/...

Question

If $w = cos (\frac{π}{n}) + i sin (\frac{π}{n})$ , then value of $1 + w + w^{2} + \dots + w^{n - 1}$ is

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$1 + i cot \frac{π}{2 n}$

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$1 + i tan \frac{π}{2 n}$

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$1 + i cos \frac{π}{2 n}$

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Solution

The correct option is B
$1 + i cot \frac{π}{2 n}$

To find the value of $1 + w + w^{2} + \dots + w^{n - 1}$

This is a Geometric Progression with first term as 1 and and common ratio as $w$

$\Rightarrow$ Sum $= \frac{1 - w^{n}}{1 - w} = \frac{1 - (cos π + i sin π)}{1 - cos \frac{π}{n} - i sin \frac{π}{n}}$

$= \frac{2}{1 - 1 + 2 {sin}^{2} \frac{π}{2 n} - 2 i sin \frac{π}{2 n} cos \frac{π}{2 n}}$ [Using $cos 2 θ = 1 - 2 {sin}^{2} θ$ and $sin 2 θ = 2 sin θ cos θ$ ]

$= \frac{2}{2 sin \frac{π}{2 n} (sin \frac{π}{2 n} - i cos \frac{π}{2 n})}$

$= \frac{sin \frac{π}{2 n} + i cos \frac{π}{2 n}}{sin \frac{π}{2 n} (sin \frac{π}{2 n} - i cos \frac{π}{2 n}) \times (sin \frac{π}{2 n} + i cos \frac{π}{2 n})}$ [Multiplying and dividing by $(sin \frac{π}{2 n} + i cos \frac{π}{2 n})$ ]

$= \frac{sin \frac{π}{2 n} + i cos \frac{π}{2 n}}{sin \frac{π}{2 n} ({sin}^{2} \frac{π}{2 n} + {cos}^{2} \frac{π}{2 n})}$

$= \frac{sin \frac{π}{2 n} + i cos \frac{π}{2 n}}{sin \frac{π}{2 n}}$

$= 1 + i cot \frac{π}{2 n}$

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