Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 13"
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== Problem == | == Problem == | ||
+ | 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) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </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> |
== Solution == | == Solution == | ||
+ | Expanding <math>\sin{(x+y)}</math> and <math>\sin{(x-y)}</math>, we have: | ||
+ | <math>(1)</math> <math>\sin{x}\cos{y}+\sin{y}\cos{x}=.3</math> | ||
+ | <math>(2)</math> <math>\sin{x}\cos{y}-\sin{y}\cos{x}=.5</math> | ||
− | = | + | <math>(1)+(2)</math> yields <math>2\sin{x}\cos{y}=.8</math> and our answer is <math>.4</math>. |
− | * [[University of South Carolina High School Math Contest/1993 Exam]] | + | |
+ | ---- | ||
+ | |||
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 12|Previous Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 14|Next Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]] | ||
+ | |||
+ | [[Category:Intermediate Trigonometry Problems]] |
Latest revision as of 12:22, 23 July 2006
Problem
Suppose that and are numbers such that and . Then
Solution
Expanding and , we have:
yields and our answer is .