Angle a lies in the second quadran and angle b lies in the third quadrant such that cos a = -3/5 and tan b = 24/7. Determine an exact value for cos (a+b), sin(a-b)

sin a = 4/5

sin b = -24/25
cos b = -7/25

cos(a+b) = cosa*cosb-sina*sinb
= (-3/5)(-7/25) - (4/5)(-24/25)
= 21/125 + 96/125
= 117/125

now apply the formula for sin(a-b) the same way.

To find the exact value of cos(a + b), and sin(a - b), we need to use trigonometric identities and formulas.

1. Finding cos(a + b):
We can use the sum of angles identity for cosine:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

First, we need to find sin(a) and sin(b), so let's determine the values.

Since angle a lies in the second quadrant and cos(a) = -3/5, we can use the Pythagorean identity:
sin^2(a) + cos^2(a) = 1

Replacing cos(a) we have:
sin^2(a) + (-3/5)^2 = 1
sin^2(a) + 9/25 = 1
sin^2(a) = 1 - 9/25
sin^2(a) = 16/25
sin(a) = ±√(16/25)
sin(a) = ±4/5

Since angle a lies in the second quadrant, sin(a) is positive.
Therefore, sin(a) = 4/5.

Next, let's find sin(b) using the given information that tan(b) = 24/7.

We can use the identity: sin(b) = tan(b)/√(1 + tan^2(b))
sin(b) = (24/7) / √(1 + (24/7)^2)
sin(b) = (24/7) / √(1 + 576/49)
sin(b) = (24/7) / √(625/49)
sin(b) = (24/7) * (√(49)/√(625))
sin(b) = (24/7) * (7/25)
sin(b) = 24/25

Now that we have sin(a) and sin(b), we can use the sum of angles identity for cosine:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a + b) = (-3/5)(-24/25) - (4/5)(24/25)
cos(a + b) = (72/125) - (96/125)
cos(a + b) = -24/125

Therefore, the exact value of cos(a + b) is -24/125.

2. Finding sin(a - b):
We can use the difference of angles identity for sine:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Using the values of sin(a) = 4/5, cos(a) = -3/5, and sin(b) = 24/25:
sin(a - b) = (4/5)(-24/25) - (-3/5)(24/25)
sin(a - b) = (-96/125) - (72/125)
sin(a - b) = -168/125

Therefore, the exact value of sin(a - b) is -168/125.

To summarize:
cos(a + b) = -24/125
sin(a - b) = -168/125