• # question_answer If $\frac{3+i\sin \theta }{4-i\cos \theta },\,\theta \in [0,\,2\pi ]$, is a real number, then an argument of $\sin \,\theta +\,i\cos \theta$ [JEE MAIN Held on 07-01-2020 Evening] A) $\pi -{{\tan }^{-1}}\left( \frac{3}{4} \right)$ B) $\pi -{{\tan }^{-1}}\left( \frac{4}{3} \right)$ C) $-{{\tan }^{-1}}\left( \frac{3}{4} \right)$   D) ${{\tan }^{-1}}\left( \frac{4}{3} \right)$

Solution :

 [b] $\because \,\,\,Z=\frac{3+i\sin \,\theta }{4-i\cos \theta }\times \frac{4+i\cos \,\theta }{4+i\cos \,\theta }$ $=\frac{(12\,-sin\theta \,\cos \theta )+i(4\,sin\theta \,+3\cos \theta )}{16+{{\cos }^{2}}\theta }$ $\because$  Z is purely real $\therefore \,\,4\sin \,\theta +3\,\cos \,\theta =0$ $\tan \theta =-\frac{3}{4}$ If$\theta \,\in \left( \frac{\pi }{2},\pi \right)$, then arg $(sin\,\theta +i\cos \theta )\,=\pi \,{{\tan }^{-1}}\left( \frac{4}{3} \right)$ if$\theta \in \left( \frac{3\pi }{2},2\pi \right)$,  then arg  $(sin\,\theta +i\cos \theta )=-\,ta{{n}^{-1}}\,\frac{4}{3}$

You need to login to perform this action.
You will be redirected in 3 sec