Prove the following

Question:

If $\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi]$, is a real number, then

an argument of $\sin \theta+i \cos \theta$ is :

  1. $-\tan ^{-1}\left(\frac{3}{4}\right)$

  2. $\tan ^{-1}\left(\frac{4}{3}\right)$

  3. $\pi-\tan ^{-1}\left(\frac{4}{3}\right)$

  4. $\pi-\tan ^{-1}\left(\frac{3}{4}\right)$


Correct Option: , 3

Solution:

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