Calculate tg(a + b), knowing that a belongs to (0 , pi/2) and belongs to (pi/2, pi). sin a = 1/2 ; sin b = 2/3.

First of all, we have to establish the signature of tg a and tg b. Due to the facts from hypothesis, tg a belongs to the first quadrant and it has the plus signature and tg b belongs to the second quadrant and it has minus signature.

tg a=sina a/cos a

cos a = (1 - sin^2a)^1/2

cos a = (1-1/4)^1/2=(3)^1/2]/2

cos b = - (1-4/9)^1/2=[-(5)^1/2]/3

tga = (1/2)/(3)^1/2]/2

tg b = (2/3)/[-(5)^1/2]/3=[-2(5)^1/2]/3

tg (a+b)=(tga +tgb)/(1-tga*tgb)

tg (a+b)={(1/2)/(3)^1/2]/2 + [-2(5)^1/2]/3}/{1 - (1/2)/(3)^1/2]/2*[-2(5)^1/2]/3}

tg (a+b)= [5(3)^1/2 - 6 (5)^1/2]/[15 + 2 (15)^1/2]