AoPSWiki

Try our innovative online adaptive learning system, Alcumus.
Over 1100 problems and 60+ video lessons. FREE!

Personal tools

Search

Toolbox

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

From AoPSWiki

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 $.4$ .

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problem_13"

Category: Intermediate Trigonometry Problems

Trying to get to the USAMO in 2010? Our AIME Problem Series can help you get there! Click here to enroll today!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us