Question 1081528
{{{  drawing( 400, 300, -20, 20, -20, 20,
  triangle( 0, 0, 8, 0, 8,15), 
  locate( 1,2,x), 
  locate( 4.5,-1, a),
  locate( 4,11.5,c),
  locate(8.5, 7, D),
  locate(9,10, b),   
  line(0,0, 8, 5),
  locate(7.2, 4, h),
  locate(4,6,d),
 locate( -2,0,A), 
  locate( 9,0,B), 
  locate( 9,15,C)

 )
}}}

Note:  In the figure  b = |BC|,  h=|BD|

{{{ sin(x) = 15/17 = b/c }}}  —>   set  b=15 and c=17  

Solve for a:    {{{ a=sqrt(c^2 - b^2) = sqrt(17^2-15^2) = sqrt(289-225) = sqrt(64) = 8 }}}

—

Because x is bisected:   {{{ (b-h)/h = c/a }}}  <<—<<< angle bisector theorem

               {{{  (15-h)/h = 17/8 }}}  —>   {{{ 25h = 120 }}} —> {{{ h=24/5 }}}

—
We need 'd':
{{{ d = sqrt(h^2 + a^2) = sqrt((24/5)^2 + 8^2) = sqrt((576/25)+64(25/25)) = (8/5)*sqrt(34) }}}

Finally, ready to solve:
{{{ sin(x/2) = h/d = (24/5) / ((8/5)*sqrt(34))   = 24/(8*sqrt(34)) = 3/sqrt(34) = (3/34)*sqrt(34) }}}

{{{ highlight(sin(x/2) = (3/34)*sqrt(34)) }}}

—

I will leave it to you to find cos(x/2).  You can use  {{{ cos(x/2) = a/d }}}  and plug in a & d (easiest way).