JEE Mains · Maths · STD 11 - 4.1 complex nubers
If the real part of the complex number \((1-\cos \theta+2 i \sin \theta)^{-1}\) is \(\frac{1}{5}\) for \(\theta \in(0, \pi)\), then the value of the integral \(\int_{0}^{\theta} \sin x \,d x\) is equal to:
- A \(2\)
- B \(-1\)
- C \(0\)
- D \(1\)
Answer & Solution
Correct Answer
(D) \(1\)
Step-by-step Solution
Detailed explanation
\(z=\frac{1}{1-\cos \theta+2 i \sin \theta}\) \(=\frac{2 \sin ^{2} \frac{\theta}{2}-2 i \sin \theta}{(1-\cos \theta)^{2}+4 \sin ^{2} \theta}\)…
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