Question 949025
Correct track, up to showing the formula, the half-angle formula for cosine.
The cosine is  {{{-3/5}}}.  Quadrant  3, according to part of your description.


Because quadrant 3,  cosine(x/2) is NEGATIVE.
{{{cos(x/2)=-sqrt ( 1 + -3/5)/2}}}


The algebra to do is very elementary.


{{{-sqrt(5/5-3/5)/2}}}
{{{-sqrt(2/5)/2}}}
{{{-(1/2)sqrt(2)/(sqrt(5))}}}
RATIONALIZE THE DENOMINATOR.
{{{-(1/2)(sqrt(2)/sqrt(5))(sqrt(5)/sqrt(5))}}}
{{{-(1/2)(sqrt(2)sqrt(5))/5}}}
{{{highlight(-sqrt(10)/10)}}}