Question 1007340
If {{{sin(a) = 8/9}}}, then {{{cos(a) = (sqrt(17))/(9)}}}. So you are correct there.


-----------------------------------------------------------------------

Start with {{{pi/2 < a < pi}}}. 


Multiply every side by {{{1/2}}} to get {{{pi/4 < a/2 < pi/2}}}


The angle {{{a/2}}} is larger than {{{pi/4}}} but smaller than {{{pi/2}}}. This places angle {{{a/2}}} in quadrant 1. So {{{sin(a/2)}}} is positive. So we only use the "plus" version of the formula (NOT the minus, NOT the plus/minus)



{{{sin(a/2) = sqrt((1-cos(a))/2)}}}


{{{sin(a/2) = sqrt((1-(sqrt(17))/(9))/2)}}}


{{{sin(a/2) = sqrt((9/9-(sqrt(17))/(9))/2)}}}


{{{sin(a/2) = sqrt(((9-sqrt(17))/(9))/2)}}}


{{{sin(a/2) = sqrt(((9-sqrt(17))/(9))*(1/2))}}}


{{{sin(a/2) = sqrt((9-sqrt(17))/(18))}}} Exact value


{{{sin(a/2) = 0.52051760426961}}} Approximate value (use a calculator)