Answer:
[tex]\frac{\sqrt{21} }{6}[/tex]
Step-by-step explanation:
Using the half angle identity
cos([tex]\frac{1}{2}[/tex] x ) = ± [tex]\sqrt{\frac{1+cosx}{2} }[/tex]
Given cosx = [tex]\frac{1}{6}[/tex] , then
cos([tex]\frac{1}{2}[/tex] x )
= ± [tex]\sqrt{\frac{1+\frac{1}{6} }{2} }[/tex]
= ± [tex]\sqrt{\frac{\frac{7}{6} }{2} }[/tex]
= ± [tex]\sqrt{\frac{7}{12} }[/tex]
= ± [tex]\frac{\sqrt{7} }{\sqrt{12} }[/tex] × [tex]\frac{\sqrt{12} }{\sqrt{12} }[/tex] ( rationalising the denominator )
= ± [tex]\frac{\sqrt{84} }{12}[/tex]
= ± [tex]\frac{2\sqrt{21} }{12}[/tex]
= [tex]\frac{\sqrt{21} }{6}[/tex] ← positive value
