A) \[255\]
B) \[265\]
C) \[275\]
D) \[770\]
Correct Answer: A
Solution :
Given, in \[\Delta \,ABC,\]\[\sin A=\frac{5}{13}\] \[\sin B=\frac{99}{101}\] We know that, \[A+B+C={{180}^{o}}\] \[\Rightarrow \] \[C={{180}^{o}}-(A+B)\] \[\Rightarrow \] \[\cos C=\cos ({{180}^{o}}-(A+B))\] \[=-\cos \,(A+B)\] \[=-[\cos \,\,A\,\,cos\,\,B-sin\,\,A\,\,\,sin\,\,B]\] \[=-[\sqrt{1-{{\sin }^{2}}A}\,\,\sqrt{1-{{\sin }^{2}}B}-\sin A\,\,\sin B]\] \[=-\left[ \sqrt{1-{{\left( \frac{5}{13} \right)}^{2}}}\,\,\sqrt{1-{{\left( \frac{99}{101} \right)}^{2}}}-\frac{5}{13}\times \frac{99}{101} \right]\] \[=-\left[ \frac{12}{13}\times \frac{20}{101}-\frac{495}{13\times 101} \right]\] \[=-\left[ \frac{240}{1313}-\frac{495}{1313} \right]=-\left[ -\left( \frac{255}{1313} \right) \right]\] \[\Rightarrow \]\[\cos \,C=\frac{255}{1313}\] Now, value of \[1313\,\,\cos \,\,(C)\] \[=1313\times \frac{255}{1313}=255\]
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