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4-1.Complex numbers
hard
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
$\pi-\tan ^{-1}\left(\frac{3}{4}\right)$
(JEE MAIN-2020)
Solution
$\frac{3+i \sin \theta}{4-i \cos \theta}$ is a real number
$\Rightarrow 3 \cos \theta+4 \sin \theta=0$
$\Rightarrow \tan \theta=\frac{-3}{4}$
argument of $\sin \theta+i \cos \theta=\pi-\tan ^{-1} \frac{4}{3}$
Standard 11
Mathematics
