Express the following complex number in polar form and exponential form: $(1 - c o s θ + i s i n θ), θ ϵ (0, π)$

2 s i n (\frac{θ}{2}) (c o s (\frac{π}{2} + \frac{θ}{2}) + i s i n (\frac{π}{2} + \frac{θ}{2})); 2 s i n (\frac{θ}{2}) e^{i (\frac{π}{2} + \frac{θ}{2})}

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- 2 s i n (\frac{θ}{2}) (c o s (\frac{π}{2} - \frac{θ}{2}) + i s i n (\frac{π}{2} - \frac{θ}{2})); - 2 s i n (\frac{θ}{2}) e^{i (\frac{π}{2} - \frac{θ}{2})}

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2 s i n (\frac{θ}{2}) (c o s (\frac{π}{2} - \frac{θ}{2}) - i s i n (\frac{π}{2} - \frac{θ}{2})); 2 s i n (\frac{θ}{2}) e^{i (\frac{π}{2} - \frac{θ}{2})}

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2 s i n (\frac{θ}{2}) (c o s (\frac{π}{2} - \frac{θ}{2}) + i s i n (\frac{π}{2} - \frac{θ}{2})); 2 s i n (\frac{θ}{2}) e^{i (\frac{π}{2} - \frac{θ}{2})}

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Solution

The correct option is C $2 s i n (\frac{θ}{2}) (c o s (\frac{π}{2} - \frac{θ}{2}) + i s i n (\frac{π}{2} - \frac{θ}{2})); 2 s i n (\frac{θ}{2}) e^{i (\frac{π}{2} - \frac{θ}{2})}$
$(1 - cos θ + i sin θ), θ \in (0, π)$
$\Rightarrow 1 - cos θ + i sin θ = x (cos θ + i sin θ)$
$\Rightarrow tan θ = \frac{sin θ}{1 - cos θ} = \frac{2 sin \frac{θ}{2} cos \frac{θ}{2}}{2 {sin}^{2} \frac{θ}{2}}$
$= cot \frac{θ}{2} \Rightarrow θ = {tan}^{- 1} (cot \frac{θ}{2})$
$∴ θ = \frac{π}{2} - \frac{θ}{2}$
$\Rightarrow r = \sqrt{{(1 - cos θ)}^{2} + {sin}^{2} θ} = \sqrt{2 - 2 cos θ} = 2 sin \frac{θ}{2}$
$\Rightarrow 1 - cos θ + i sin θ = 2 sin \frac{θ}{2} e^{(π / 2 - θ / 2) i}$
$\Rightarrow 2 sin (\frac{θ}{2}) [cos (\frac{π}{2} - \frac{θ}{2}) + i sin (\frac{π}{2} - \frac{θ}{2})]; 2 sin \frac{θ}{2} e^{(π / 2 - θ / 2) i}$
Hence, the answer is $2 sin (\frac{θ}{2}) [cos (\frac{π}{2} - \frac{θ}{2}) + i sin (\frac{π}{2} - \frac{θ}{2})]; 2 sin (\frac{θ}{2}) e^{(π / 2 - θ / 2) i} .$

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