question_answer

Find \[\cos \,(x/2),\] if \[\tan \,x=5/12,\,x\] in quadrant III

A)  \[5/\sqrt{13}\]            

B)  \[5/\sqrt{26}\]

C)  \[5/13\]              

D)  \[5/26\]

E)  None of these

Correct Answer: E

Solution :

Given,  \[\tan x=\frac{5}{12}\] and  x is in II quadrant. \[\therefore \] \[\sin x=\frac{-5}{13}\] and \[\cos x=\frac{-12}{13}\] Now, \[\cos x=2{{\cos }^{2}}\frac{x}{2}-1\] \[\Rightarrow \] \[{{\cos }^{2}}\frac{x}{2}=\frac{1}{2}(\cos x+1)\] \[=\frac{1}{2}\left( \frac{-12}{13}+1 \right)\] \[=\frac{1}{2}\left( \frac{1}{13} \right)=\frac{1}{26}\,\,\Rightarrow \cos \frac{x}{2}=\sqrt{\frac{1}{26}}\]