Problem:
Suppose that sina+sinb=5/3 and cosa+cosb=1. What is cos(a−b)?
Answer Choices:
A. 35−1
B. 31
C. 21
D. 32
E. 1
Solution:
Square both sides of both given equations to obtain
sin2a+2sinasinb+sin2b=5/3 and cos2a+2cosacosb+cos2b=1
Then add corresponding sides of the resulting equations to obtain
(sin2a+cos2a)+(sin2b+cos2b)+2(sinasinb+cosacosb)=38.
Because sin2a+cos2a=sin2b+cos2b=1, it follows that
cos(a−b)=sinasinb+cosacosb=31.
One ordered pair (a,b) that satisfies the given condition is approximately (0.296,1.527).
The problems on this page are the property of the MAA's American Mathematics Competitions