43) Angle a lies in the second quadrant and angle b lies in the third quadrant such that cos a = −
3
5
and tan b =
24
7
. Determine an exact value for
diagrams:
a) cos(a + b)
To find cos(a + b), we can use the formula:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
We are given that cos(a) = -3/5 and tan(b) = 24/7. We can use this information to find sin(a) and cos(b).
Since cos^2(a) + sin^2(a) = 1, we can solve for sin(a):
(-3/5)^2 + sin^2(a) = 1
9/25 + sin^2(a) = 1
sin^2(a) = 1 - 9/25
sin^2(a) = 16/25
sin(a) = ±√(16/25)
sin(a) = ±4/5
Since tan(b) = sin(b)/cos(b), we can use this to solve for cos(b):
tan(b) = sin(b)/cos(b)
24/7 = sin(b)/cos(b)
24cos(b) = 7sin(b)
24cos(b) - 7sin(b) = 0
Using the identity cos(90 - x) = sin(x), we can rewrite this as:
24cos(b) - 7sin(90 - b) = 0
24cos(b) - 7cos(b) = 0
17cos(b) = 0
cos(b) = 0
Now let's substitute these values into the formula for cos(a + b):
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a + b) = (-3/5)(0) - (4/5)(0) (since cos(b) = 0 and sin(b) = 0)
cos(a + b) = 0
Therefore, cos(a + b) is equal to 0.