How do I find the value of sin(a-b) if tana=4/3, cotb=5/12, 0<a<90, and 0<b<90?

Answer 1

so both a and b are in the first quadrant.

Draw two right-angled triangles in the first quadrant with
1. angle a and opposite of 4 and adjacent of 3 (tan a=4/3)
then the hypotenuse is 5 and
sin a = 4/5, cos a = 3/5
2. angle b and opposite 12 and adjacent of 5 (cot b = 5/12)
then the hypotenuse is 13 and
sin b = 12/13, cos b = 5/13

we know
sin(a - b)
= sinacosb - cosasinb
= (4/5)(5/13) - (3/5)(12/13)
= -16/65

Answer 2

I was only two steps from being right!