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University of South Carolina High School Math Contest/1993 Exam/Problem 13

Problem

Suppose that $x$ and $y$ are numbers such that $\sin(x+y) = 0.3$ and $\sin(x-y) = 0.5$. Then $\sin (x)\cdot \cos (y) =$

$\mathrm{(A) \ }0.1 \qquad \mathrm{(B) \ }0.3 \qquad \mathrm{(C) \ }0.4 \qquad \mathrm{(D) \ }0.5 \qquad \mathrm{(E) \ }0.6$

Solution

Expanding $\sin{(x+y)}$ and $\sin{(x-y)}$, we have: $(1)$ $\sin{x}\cos{y}+\sin{y}\cos{x}=.3$ $(2)$ $\sin{x}\cos{y}-\sin{y}\cos{x}=.5$

$(1)+(2)$ yields $2\sin{x}\cos{y}=.8$ and our answer is $.4$.

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