Question

I need help with this trig practice problem Having trouble with it

Answer 1

We want to evaluate

`\begin{gathered} \cos ((x)/(2)) \\ \text{If }\cos (x)=-(2)/(5)\text{ and in the third quadrant } \end{gathered}`

Using the half angle formula, we have

`\begin{gathered} \cos ((x)/(2))=\pm\sqrt[]{(1+\cos(x))/(2)} \\ =\pm\sqrt[]{(1-(2)/(5))/(2)} \\ =\pm\sqrt[]{((3)/(5))/(2)} \\ =\pm\sqrt[]{(3)/(10)} \end{gathered}`

Now since the angle (x) is the third quadrant, that means

`\begin{gathered} (x)/(2)\text{ should fall under the second quadrant } \\ \text{and in the second quadrant, } \\ cos\theta\text{ is negative} \end{gathered}`

Hence the answer becomes

`\cos ((x)/(2))=-\sqrt[]{(3)/(10)}`