Use the given information to evaluate cos (a-b).
Cot a = 3/4, cot b = 24/25; pi<a<3pi/2, 3pi/2<b<2pi.
since a is in QIII,
sina = -4/5
cosa = -3/5
You sure that's not cos b = 24/25? cot < 0 in QIV. If so, then we have a 7-24-25 triangle.
sinb = 7/25
cosb = 24/25
now just use your difference formula:
cos(a-b) = cosa cosb + sina sinb
To evaluate cos(a-b), we need to use the trigonometric identity:
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
First, let's find the values of sin(a) and sin(b).
Given: cot(a) = 3/4
Since cot is the reciprocal of tan, we can write:
cot(a) = 3/4 = cos(a)/sin(a)
We can solve this equation for sin(a):
sin(a) = cos(a) / cot(a)
sin(a) = cos(a) / (3/4)
sin(a) = (4/3)cos(a)
Similarly, given: cot(b) = 24/25
cot(b) = 24/25 = cos(b)/sin(b)
Solving for sin(b):
sin(b) = cos(b) / cot(b)
sin(b) = cos(b) / (24/25)
sin(b) = (25/24)cos(b)
Now we have sin(a) and sin(b), and we can evaluate cos(a-b) using the trigonometric identity:
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + [(4/3)cos(a)][(25/24)cos(b)]
Knowing that 𝜋 < a < (3𝜋)/2 and (3𝜋)/2 < b < 2𝜋, we can now substitute cos(a) and cos(b) with their corresponding values.
However, the values of cos(a) and cos(b) are currently unknown. So, we need further information to evaluate cos(a-b) accurately.