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) \[{{\tan }^{-1}}\left( \frac{4}{3} \right)\]
Correct Answer: B
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 \[(sin\,\theta +i\cos \theta )=-\,ta{{n}^{-1}}\,\frac{4}{3}\] |
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