can someone help explain how to find the exact values of the expressions without solving for x

Question

If tan(x) = -3/4 and x is in quadrant IV, find the exact values of the expressions without solving for x

sin(x/2) =

cos(x/2)=

tan(x/2)=

Answer 1

tan(x) = -3/4

tan^2(x) + 1 = sec^2(x) = 25/16

sec(x) = +/- 5/4

cos(x) = 1/sec(x) = 4/5 (since in quadrant 4)

sin(x) = 3/5

sin(x/2) = +/- sqrt((1-cosx)/2) = sqrt((1-4/5)/2) = +/- 0.1

cos(x/2) = +/- sqrt((1+cosx)/2) = sqrt((1+4/5)/2) = +- 0.9

tan(x/2) = sinx / (1 + cosx) = 3/5 / (1 + 4/5) = 0.33

Answer 2

tan = -3 / 4 in QIV → sin x = - 3 / √17, and cos x = 4 / √17

Now use the half-angle formulas to answer.