Question

if sin a equal to 1 by 2 cos B equal to 12 by 13 where pi by 2 is greater than A is greater than pi and 3 pi by 2 is greater than B is greater than 2 find tan a minus b ​

Answer 1

Told you already .

sin A = 1/2

since π/2 < A < π ( A is in second quadrant )

TanA =(-1/√3 ) -----------(1)

now, given CosB = 12/13

and , 3π/2 < B < 2π (B is in 4th quadrant)

TanB = (-5/12) -----------(2)

put values of (1) and (2) in formula , we get

Tan(A-B) = (-1/√3) - (-5/12) /1+(-1/√3)*(-5/12)

= (5/12-1/√3)/(1 + 5/12√3)

= (5√3-12)/(12√3+5)

rationalize this further if you want ,

(5√3-12)(12√3-5)/(144*3-25)

(180-25√3-144√3+60)/407

(240-119√3)/407 (ans.)