The exact value of cos (x/2) located in the third quadrant is determined as [tex]-\frac{2}{\sqrt{5} }[/tex].
Angle x in the third quadrant
The value of angle x in the third quadrant can be determined by applying trignometary ratio as follows;
[tex]Cos(\frac{x}{2} ) = \pm\sqrt{\frac{1 + cos (x)}{2}} \\\\[/tex]
for the given value of cos x = -1/5, the value of cos(x/2) is calculated as follows;
[tex]Cos(\frac{x}{2} ) = \pm\sqrt{\frac{1 + (\frac{-1}{5} )}{2}} \\\\Cos(\frac{x}{2} ) = \pm\sqrt{\frac{\frac{4}{5} }{2}}\\\\Cos(\frac{x}{2} ) =\pm\sqrt{\frac{4 }{10}}\\\\Cos(\frac{x}{2} ) =\pm\frac{2}{\sqrt{5} }[/tex]
Thus, the exact value of cos (x/2) located in the third quadrant is determined as [tex]-\frac{2}{\sqrt{5} }[/tex].
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