Suppose that sin a+sin b=√53 and cos a+cos b=1. What is cos(a-b)?

2007 AMC 12A Problem 17

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17.

Suppose that

\sin a+\sin b=\sqrt{\tfrac53}

and

\cos a+\cos b=1.

What is

\cos(a-b)?

\sqrt{\dfrac53}-1

\dfrac13

\dfrac12

\dfrac23

1

Square both given equations and add them

Use

\sin^2\theta+\cos^2\theta=1

to isolate

\sin a\sin b+\cos a\cos b

Squaring both equations gives

\sin^2 a+2\sin a\sin b+\sin^2 b=\tfrac53

and

\cos^2 a+2\cos a\cos b

+\cos^2 b=1.

Adding and using

\sin^2\theta+\cos^2\theta=1

twice,

\begin{aligned} &2+2(\sin a\sin b+\cos a\cos b) \\ &=\tfrac83. \end{aligned}

\cos(a-b)=\sin a\sin b

+\cos a\cos b

=\tfrac13.

Thus, the correct answer is B.