A) \[\sqrt{6}\]
B) \[36\]
C) \[6\]
D) None of these
Correct Answer: D
Solution :
Given that, \[a=5,\text{ }b=4\]and \[\cos \,(A+B)=\frac{31}{32}\] \[\Rightarrow \] \[\cos \,(\pi -C)=\frac{31}{32}\] \[\Rightarrow \] \[-\cos \,C=-\frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}=\frac{31}{32}\] \[\Rightarrow \] \[\frac{{{(5)}^{2}}+{{(4)}^{2}}-{{c}^{2}}}{2\times 5\times 4}=-\frac{31}{32}\] \[\Rightarrow \] \[41-{{c}^{2}}=-\frac{155}{4}\] \[\Rightarrow \] \[{{c}^{2}}=41+\frac{155}{4}\] \[\Rightarrow \] \[{{c}^{2}}=\frac{319}{4}\] \[\Rightarrow \] \[c=\sqrt{\frac{319}{2}}\]
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