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

(Created page with "==Problem== If <math>\sin(x) =3 \cos(x) </math> then what is <math>\sin(x)\cos(x)</math>? <math>\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{5}\qquad \textbf{(C)}\ \f...")
 
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==Problem==
 
==Problem==
  
−
If <math>\sin(x) =3 \cos(x) </math> then what is <math>\sin(x)\cos(x)</math>?
+
If <math>\sin(x) =3 \cos(x) </math> then what is <math>\sin(x) \cdot \cos(x)</math>?
  
 
<math>\textbf{(A)}\ \frac{1}{6}\qquad
 
<math>\textbf{(A)}\ \frac{1}{6}\qquad
Line 7: Line 7:
 
\textbf{(C)}\ \frac{2}{9}\qquad
 
\textbf{(C)}\ \frac{2}{9}\qquad
 
\textbf{(D)}\ \frac{1}{4}\qquad
 
\textbf{(D)}\ \frac{1}{4}\qquad
−
\textbf{(E)}\ \frac{3}{10} </math>  
+
\textbf{(E)}\ \frac{3}{10} </math>
  
 
==Solution==
 
==Solution==
 +
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>
 +
We can now use right triangle trigonometry to finish the problem.
 +
<asy>
 +
pair A,B,C;
 +
A = (0,0);
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B = (3,0);
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C = (0,1);
 +
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);
 +
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);
 +
</asy>
  
 +
(Note that this assumes that <math>x</math> is acute. If <math>x</math> is obtuse, then <math>\sin{(x)}</math> is positive and <math>\cos{(x)}</math> is negative, so the equation cannot be satisfied. If <math>x</math> is reflex, then both <math>\sin{(x)}</math> and <math>\cos{(x)}</math> are negative, so the equation is satisfied, but when we find <math>\sin{(x)}\cos{(x)}</math>, the two negatives will cancel out and give the same (positive) answer as in the acute case.)
  
 +
Since the problem asks us to find <math>\sin{(x)}\cos{(x)}</math>.
 +
<cmath>\sin{(x)}\cos{(x)}=\left(\frac{3}{\sqrt{10}}\right)\left(\frac{1}{\sqrt{10}}\right)=\frac{3}{10}.</cmath>
 +
So <math>\boxed{\textbf{(E)}\ \frac{3}{10}}</math> is our answer.
 +
 +
==Solution 2 (Pure Algebra)==
 +
 +
Squaring both sides gives <math>{\sin}^2 x = 9{\cos}^2 x</math>. We can take the Pythagorean identity, <math>{\sin}^2 x + {\cos}^2 x = 1</math> and substitute the 1st equation in, giving <math>10{\cos}^2 x = 1</math>. So <math>{\cos}^2 x = \frac{1}{10}</math>, and <math>{\sin}^2 x = \frac{9}{10}</math>.
 +
 +
Multiplying the 2 together gives <math>{\sin}^2 x {\cos}^2 x = \frac{9}{100}</math>, and then taking the square root gives <math>\mp \frac{3}{10}</math>. However, <math>-\frac{3}{10}</math> implies one of <math>\sin x</math> and <math>\cos x</math> is negative while the other is positive, but <math>\sin x = 3 \cos x</math> means they have the same sign, which contradicts the first statement. This means <math>\boxed{\textbf{(E)}\ \frac{3}{10}}</math> is the only answer.
 +
 +
-ThisUsernameIsTaken
  
 
== See also ==
 
== See also ==

Latest revision as of 19:53, 24 August 2021

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]

(Note that this assumes that $x$ is acute. If $x$ is obtuse, then $\sin{(x)}$ is positive and $\cos{(x)}$ is negative, so the equation cannot be satisfied. If $x$ is reflex, then both $\sin{(x)}$ and $\cos{(x)}$ are negative, so the equation is satisfied, but when we find $\sin{(x)}\cos{(x)}$, the two negatives will cancel out and give the same (positive) answer as in the acute case.)

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

Solution 2 (Pure Algebra)

Squaring both sides gives ${\sin}^2 x = 9{\cos}^2 x$. We can take the Pythagorean identity, ${\sin}^2 x + {\cos}^2 x = 1$ and substitute the 1st equation in, giving $10{\cos}^2 x = 1$. So ${\cos}^2 x = \frac{1}{10}$, and ${\sin}^2 x = \frac{9}{10}$.

Multiplying the 2 together gives ${\sin}^2 x {\cos}^2 x = \frac{9}{100}$, and then taking the square root gives $\mp \frac{3}{10}$. However, $-\frac{3}{10}$ implies one of $\sin x$ and $\cos x$ is negative while the other is positive, but $\sin x = 3 \cos x$ means they have the same sign, which contradicts the first statement. This means $\boxed{\textbf{(E)}\ \frac{3}{10}}$ is the only answer.

-ThisUsernameIsTaken

See also

1988 AHSME (Problems • Answer Key • Resources)
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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