question_answer

In a \[\Delta ABC,\] it is given that \[\angle C={{90}^{o}}\] and \[\tan A=\frac{1}{\sqrt{3}},\]find the value of \[(\sin A\,\cos B+\cos A\,\sin B)\].

A) 1                                 

B) \[\frac{1}{2}\]            

C)  0                    

D)         3

Correct Answer: A

Solution :

Consider \[\Delta ABC\] in which \[\angle C={{90}^{o}}\] and \[\tan A=\frac{1}{\sqrt{3}}.\] Let \[BC=x.\] Then,   \[AC=\sqrt{3}x\] By Pythagoras' theorem, we have, \[A{{B}^{2}}=A{{C}^{2}}+B{{C}^{2}}={{(\sqrt{3}x)}^{2}}+{{x}^{2}}=2x\]  \[\therefore \] \[\sin A=\frac{x}{2x}=\frac{1}{2}\] and \[\cos A=\frac{\sqrt{3}x}{2x}=\frac{\sqrt{3}}{2}\] and \[\sin B=\frac{\sqrt{3}x}{2x}=\frac{\sqrt{3}}{2}\] and \[\cos B=\frac{x}{2x}=\frac{1}{2}\] \[\therefore \]    \[(\sin A\,\cos B+\cos A\sin B)\]             \[=\left( \frac{1}{2}\times \frac{1}{2}+\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2} \right)=\left( \frac{1}{4}+\frac{3}{4} \right)=1\]