SOLUTION: find the exact value of cos(a+B) if sin a= -4/5, and sin B= 5/13

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Question 873142: find the exact value of cos(a+B)
if sin a= -4/5, and sin B= 5/13
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
find the exact value of cos(a+B)
if sin a= -4/5, and sin B= 5/13

cos(a) = ±√1-sinČ(a) = %22%22+%2B-+sqrt%281-%28-4%2F5%29%5E2%29 = %22%22+%2B-+sqrt%281-16%2F25%29%29 = %22%22+%2B-+sqrt%2825%2F25-16%2F25%29%29 = %22%22+%2B-+sqrt%289%2F25%29%29 = %22%22+%2B-+3%2F5%29

cos(B) = ±√1-sinČ(B) = %22%22+%2B-+sqrt%281-%285%2F13%29%5E2%29 = %22%22+%2B-+sqrt%281-25%2F169%29%29 = %22%22+%2B-+sqrt%28169%2F169-25%2F169%29%29 = %22%22+%2B-+sqrt%28144%2F169%29%29 = %22%22+%2B-+12%2F13%29 

Case 1:  a is in Q3 and B is in Q1

cos(a+B) = cos(a)cos(B) - sin(a)sin(B) = (-3/5)(12/13) - (-4/5)(5/13) = -16/65

Case 2:  a is in Q3 and B is in Q2

cos(a+B) = cos(a)cos(B) - sin(a)sin(B) = (-3/5)(-12/13) - (-4/5)(5/13) = 56/65

Case 3:  a is in Q4 and B is in Q1

cos(a+B) = cos(a)cos(B) - sin(a)sin(B) = (3/5)(12/13) - (-4/5)(5/13) = 56/65

Case 4:  a is in Q4 and B is in Q2 

cos(a+B) = cos(a)cos(B) - sin(a)sin(B) = (3/5)(-12/13) - (-4/5)(5/13) = -16/65

Two possible solutions: -16/65 and 56/65

Edwin