If sin theta 45 and theta is in Quadrant III find tantheta2

If sin theta = -4/5 and theta is in Quadrant III, find tan(theta/2).

sin(theta) = - 4/ 5

theta is in III qudrant

we know sin(theta) = opp / hyp = - 4 / 5

opp = - 4 and hyp = 5

by pythogorous theorem

hyp^2 = opp^2 + adj^2

adj^2 = (-4)^2 + 5 ^2

adj^2 = 16 + 25

adj^2 = 41

adj = + sqrt(41)

In III qudrant sin,cos are negative , so adj = - sqrt(41)

cos(theta) = adj / hyp ==> - sqrt(41) / 5

we know half-angle identities

tan(theta/2 ) = [ 1 - ( - sqrt(41) / 5) ] / - 4/5

==> [ ( 5+ (sqrt(41) ) / 5) ] / - 4/5

==> ( 5+ (sqrt(41) ) / - 4

===> - ( 5+ (sqrt(41) ) / 4

==> -2.85078