A) \[4:3\]
B) \[5:3\]
C) \[3:4\]
D) \[3:5\]
Correct Answer: A
Solution :
Let \[\sin A=3k,\,\sin B=4k,\,\sin C=5k\] \[\therefore \] \[\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=p(let)\] \[\Rightarrow \] \[\frac{3k}{a}=\frac{4k}{b}=\frac{5k}{c}=p\] \[\Rightarrow \] \[a=3\left( \frac{k}{p} \right),b=4\left( \frac{k}{p} \right),c=5\left( \frac{k}{p} \right)\] \[\Rightarrow \] \[a=3l,\,\,b=4l,\,\,c=5l\] \[\left( let\,\,\,l=\frac{k}{p} \right)\] Now, \[\cos A=\frac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}=\frac{16+25-9}{2\times 4\times 5}\] \[=\frac{32}{40}=\frac{4}{5}\] Now, \[\operatorname{cosB}=\frac{{{c}^{2}}+{{a}^{2}}-{{b}^{2}}}{2ac}=\frac{25+9-16}{2\times 3\times 5}\] \[=\frac{18}{30}=\frac{3}{5}\] Now, \[\cos A:\cos B=\frac{4}{5}:\frac{3}{5}\] \[\Rightarrow \] \[\cos \,\,A:\,\,\cos B=4:3\]
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