JEE Mains · Maths · STD 11 - 4.1 complex nubers
If \(\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi],\) is a real number, then an argument of \(\sin \theta+\mathrm{i} \cos \theta\) is
- A \(-\tan ^{-1}\left(\frac{3}{4}\right)\)
- B \(\tan ^{-1}\left(\frac{4}{3}\right)\)
- C \(\pi-\tan ^{-1}\left(\frac{4}{3}\right)\)
- D \(\pi-\tan ^{-1}\left(\frac{3}{4}\right)\)
Answer & Solution
Correct Answer
(C) \(\pi-\tan ^{-1}\left(\frac{4}{3}\right)\)
Step-by-step Solution
Detailed explanation
\(\frac{3+i \sin \theta}{4-i \cos \theta}\) is a real number \(\Rightarrow 3 \cos \theta+4 \sin \theta=0\) \(\Rightarrow \tan \theta=\frac{-3}{4}\) argument of \(\sin \theta+i \cos \theta=\pi-\tan ^{-1} \frac{4}{3}\)
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