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Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 13"
 
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== Problem ==
 
== Problem ==
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Suppose that <math>x</math> and <math>y</math> are numbers such that <math>\sin(x+y) = 0.3</math> and <math>\sin(x-y) = 0.5</math>.  Then <math>\sin(x)\cos(y)=</math>
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Suppose that <math>x</math> and <math>y</math> are numbers such that <math>\sin(x+y) = 0.3</math> and <math>\sin(x-y) = 0.5</math>.  Then <math> \sin (x)\cdot \cos (y) = </math>
  
 
<center><math> \mathrm{(A) \ }0.1 \qquad \mathrm{(B) \ }0.3 \qquad \mathrm{(C) \ }0.4 \qquad \mathrm{(D) \ }0.5 \qquad \mathrm{(E) \ }0.6  </math></center>
 
<center><math> \mathrm{(A) \ }0.1 \qquad \mathrm{(B) \ }0.3 \qquad \mathrm{(C) \ }0.4 \qquad \mathrm{(D) \ }0.5 \qquad \mathrm{(E) \ }0.6  </math></center>
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<math>(1)+(2)</math> yields <math>2\sin{x}\cos{y}=.8</math> and our answer is <math>.4</math>.
 
<math>(1)+(2)</math> yields <math>2\sin{x}\cos{y}=.8</math> and our answer is <math>.4</math>.
  
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== See also ==
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* [[University of South Carolina High School Math Contest/1993 Exam]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 12|Previous Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 14|Next Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]]
  
 
[[Category:Intermediate Trigonometry Problems]]
 
[[Category:Intermediate Trigonometry Problems]]

Latest revision as of 12:22, 23 July 2006

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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