help with precalc

Question

if , and  is in quadrant II, determine the exact value of each of the following. Use fractions and radicals only, no decimals. You do not need to rationalize the denominator.

sin(x/2)=

cos(x/2)=

tan(x/2)=

Answer 1

First need to find cos(x)

sin(x) = opp/hyp = 5/9

cos(x) = adj/hyp = adj/9

adjacent side = √(9²-5²) = √56 = 2√14

cos(x) = adj/hyp = 2√14/9

Now you can use the half angle formulas

sin(x/2) = √((1-cos(x))/2) = √(1-2(√14/9)/2) = √((9-√14)/9) = √(9-√14)/3

Likewise:

cos(x/2) = √(9+√14)/3

tan(x/2) = sin(x/2)/cos(x/2) = √(9-√14)/√(9+√14) = √((9-√14)/(9+√14))