Answer:
cos(A+B) = 207/305
Explanation:
You want the simplest form of cos(A+B), where tan(A) = 11/60 and sin(B) = 3/5.
Cosine of sum
The identity for the cosine of the sum of angles is ...
cos(A+B) = cos(A)cos(B) -sin(A)sin(B)
In order to use this formula, we would need to find the sine and cosine of A, and the cosine of B.
Angle A
The two numbers in the ratio for tan(A) represent legs of a right triangle. The hypotenuse of that triangle is ...
c² = a² +b²
c² = 11² +60² = 121 +3600 = 3721
c = √3721 = 61
Then the trig values of interest are ...
- sin(A) = 11/61
- cos(A) = 60/61
Angle B
The cosine of angle B is ...
cos(B) = √(1 -sin²(B)) = √(1 -(3/5)²) = √(16/25) = 4/5
Sum
Then our cosine is ...
cos(A+B) = (60/61)(4/5) -(11/61)(3/5) = (60·4 -11·3)/(61·5)
cos(A+B) = 207/305