A) \[\frac{69}{48}\]
B) \[\frac{96}{48}\]
C) \[\frac{48}{69}\]
D) None of these
Correct Answer: A
Solution :
We have \[\frac{\sin A}{4}=\frac{\sin B}{5}=\frac{\sin C}{6}\Rightarrow \frac{a}{4}=\frac{b}{5}=\frac{c}{6}=\lambda \] (say) \[\therefore \] \[a=4\lambda ,\,b=5\lambda ,\,c=6\lambda \] Now \[\cos A+\cos B+\cos C\] = \[\frac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}+\frac{{{c}^{2}}+{{a}^{2}}-{{b}^{2}}}{2ac}+\frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}\] = \[\frac{1}{240{{\lambda }^{3}}}\{4\lambda (45{{\lambda }^{2}}+5\lambda (27{{\lambda }^{2}})+6\lambda (5{{\lambda }^{2}})\}=\frac{69}{48}\].
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