Expand sin(u - v) using the angle sum formula for sine:
sin(u - v) = sin(u) cos(v) - cos(u) sin(v)
Recall the Pythagorean identity: for all x,
cos²(x) + sin²(x) = 1
Dividing both sides by cos²(x) gives
1 + tan²(x) = sec²(x)
(since tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x))
With 3π/2 < u < 2π, we expect sin(u) < 0, and with π/2 < v < π, we expect cos(v) < 0 and sin(v) > 0.
Solve for sin(u) :
sin²(u) = 1 - (4/5)²
sin(u) = - √(1 - 16/25)
sin(u) = - √(9/25)
sin(u) = -3/5
Solve for sec(v) :
sec²(v) = 1 + (-12/5)²
sec(v) = - √(169/25)
sec(v) = -13/5
Take the reciprocal of both sides to solve for cos(v) :
1/sec(v) = 1 / (-13/5)
cos(v) = -5/13
Solve for sin(v) :
sin²(v) = 1 - cos²(v)
sin(v) = √(1 - (-5/13)²)
sin(v) = √(144/169)
sin(v) = 12/13
Now put everything together:
sin(u - v) = (-3/5) (-5/13) - (4/5) (12/13) = -33/65
