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Question: The modulus of the complex number \(\frac{(1 - i\sqrt{3})(\cos\theta + i\sin\theta)}{2(1 - i)(\cos\t...

The modulus of the complex number (1−i3)(cos⁡θ+isin⁡θ)2(1−i)(cos⁡θ−isin⁡θ)\frac{(1 - i\sqrt{3})(\cos\theta + i\sin\theta)}{2(1 - i)(\cos\theta - i\sin\theta)}is

A

12\frac{1}{\sqrt{2}}

B

122\frac{1}{2\sqrt{2}}

C

13\frac{1}{\sqrt{3}}

D

None of these

Answer

12\frac{1}{\sqrt{2}}

Explanation

Solution

Sol. ∣1−i3∣.∣cos⁡θ+isin⁡θ∣∣2∣.∣1−i∣.∣cos⁡θ−isin⁡θ∣=1+3.cos⁡2θ+sin⁡2θ21+1.cos⁡2θ+sin⁡2θ=12\frac{|1 - i\sqrt{3}|.|\cos\theta + i\sin\theta|}{|2|.|1 - i|.|\cos\theta - i\sin\theta|} = \frac{\sqrt{1 + 3}.\sqrt{\cos^{2}\theta + \sin^{2}\theta}}{2\sqrt{1 + 1}.\sqrt{\cos^{2}\theta + \sin^{2}\theta}} = \frac{1}{\sqrt{2}}