Difference between revisions of "1988 AHSME Problems/Problem 13"

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(Solution)
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==Solution==
 
==Solution==
 
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In the problem we are given that <math>\sin{(x)}=3\cos{(x)}</math>, and we want to find <math>\sin{(x)}\cos{(x)}</math>. We can divide both sides of the original equation by <math>\cos{(x)}</math> to get <cmath>\frac{\sin{(x)}}{\cos{(x)}}=\tan{(x)}=3.</cmath>
 
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We can now use right triangle trigonometry to finish the problem.
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<asy>
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pair A,B,C;
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A = (0,0);
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B = (3,0);
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C = (0,1);
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draw(A--B--C--A);
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draw(rightanglemark(B,A,C,8));
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label("$A$",A,SW);
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label("$B$",B,SE);
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label("$C$",C,N);
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label("$3$",B/2,S);
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label("$1$",C/2,W);
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label("$\sqrt{10}$",(C+B)/2,NE);
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</asy>
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 +
Since the problem asks us to find <math>\sin{(x)}\cos{(x)}</math>.
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<cmath>\sin{(x)}\cos{(x)}=\frac{3}{\sqrt{10}}\frac{1}{\sqrt{10}}=\frac{3}{10}.</cmath>
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So <math>\boxed{\textbf{(E)}\ \frac{3}{10}}</math> is our answer.
  
 
== See also ==
 
== See also ==

Revision as of 10:26, 2 January 2016

Problem

If $\sin(x) =3 \cos(x)$ then what is $\sin(x) \cdot \cos(x)$?

$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{5}\qquad \textbf{(C)}\ \frac{2}{9}\qquad \textbf{(D)}\ \frac{1}{4}\qquad \textbf{(E)}\ \frac{3}{10}$

Solution

In the problem we are given that $\sin{(x)}=3\cos{(x)}$, and we want to find $\sin{(x)}\cos{(x)}$. We can divide both sides of the original equation by $\cos{(x)}$ to get \[\frac{\sin{(x)}}{\cos{(x)}}=\tan{(x)}=3.\] We can now use right triangle trigonometry to finish the problem. [asy] pair A,B,C; A = (0,0); B = (3,0); C = (0,1); draw(A--B--C--A); draw(rightanglemark(B,A,C,8)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$3$",B/2,S); label("$1$",C/2,W); label("$\sqrt{10}$",(C+B)/2,NE); [/asy]

Since the problem asks us to find $\sin{(x)}\cos{(x)}$. \[\sin{(x)}\cos{(x)}=\frac{3}{\sqrt{10}}\frac{1}{\sqrt{10}}=\frac{3}{10}.\] So $\boxed{\textbf{(E)}\ \frac{3}{10}}$ is our answer.

See also

1988 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
All AHSME Problems and Solutions

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