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Difference between revisions of "2003 AIME I Problems/Problem 4"
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== Problem ==
 
== Problem ==
 
Given that <math> \log_{10} \sin x + \log_{10} \cos x = -1 </math> and that <math> \log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1), </math> find <math> n. </math>
 
Given that <math> \log_{10} \sin x + \log_{10} \cos x = -1 </math> and that <math> \log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1), </math> find <math> n. </math>
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== Solution ==
+
 
−
Using the properties of [[logarithm]]s, we can simplify the first equation to <math>\log_{10} \sin x + \log_{10} \cos x = \log_{10}(\sin x \cos x) = -1 </math>. Therefore, <math> \sin x \cos x = \frac{1}{10}\ (*)</math>.
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== Solution 1 ==
 +
Using the properties of [[logarithm]]s, we can simplify the first equation to <math>\log_{10} \sin x + \log_{10} \cos x = \log_{10}(\sin x \cos x) = -1 </math>. Therefore, <cmath> \sin x \cos x = \frac{1}{10}.\qquad (*)</cmath>
  
 
Now, manipulate the second equation.
 
Now, manipulate the second equation.
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<center><math>\begin{align*}
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<cmath>\begin{align*}
 
\log_{10} (\sin x + \cos x) &= \frac{1}{2}(\log_{10} n - \log_{10} 10) \\
 
\log_{10} (\sin x + \cos x) &= \frac{1}{2}(\log_{10} n - \log_{10} 10) \\
 
\log_{10} (\sin x + \cos x) &= \left(\log_{10} \sqrt{\frac{n}{10}}\right) \\
 
\log_{10} (\sin x + \cos x) &= \left(\log_{10} \sqrt{\frac{n}{10}}\right) \\
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\sin^2 x + \cos^2 x +2 \sin x \cos x &= \frac{n}{10} \\
 
\sin^2 x + \cos^2 x +2 \sin x \cos x &= \frac{n}{10} \\
 
\end{align*}
 
\end{align*}
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</math></center>
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</cmath>
  
 
By the Pythagorean identities, <math>\sin ^2 x + \cos ^2 x = 1</math>, and we can substitute the value for <math>\sin x \cos x</math> from <math>(*)</math>. <math>1 + 2\left(\frac{1}{10}\right) = \frac{n}{10} \Longrightarrow n = \boxed{012} </math>.
 
By the Pythagorean identities, <math>\sin ^2 x + \cos ^2 x = 1</math>, and we can substitute the value for <math>\sin x \cos x</math> from <math>(*)</math>. <math>1 + 2\left(\frac{1}{10}\right) = \frac{n}{10} \Longrightarrow n = \boxed{012} </math>.
 +
 +
== Solution 2 ==
 +
Examining the first equation, we simplify as the following:
 +
<cmath>\log_{10} \sin x \cos x = -1</cmath>
 +
<cmath>\implies \sin x \cos x = \frac{1}{10}</cmath>
 +
 +
With this in mind, examining the second equation, we may simplify as the following (utilizing logarithm properties):
 +
<cmath>\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - \log_{10} 10)</cmath>
 +
<cmath>\implies \log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} \frac{n}{10})</cmath>
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<cmath>\implies \log_{10} (\sin x + \cos x) = \log_{10} \sqrt{\frac{n}{10}}</cmath>
 +
 +
From here, we may divide both sides by <math>\log_{10} (\sin x + \cos x)</math> and then proceed with the change-of-base logarithm property:
 +
<cmath>1 = \frac{\log_{10} \sqrt{\frac{n}{10}}}{\log_{10} (\sin x + \cos x)}</cmath>
 +
<cmath>\implies 1 = \log_{\sin x + \cos x} \sqrt{\frac{n}{10}}</cmath>
 +
 +
Thus, exponentiating both sides results in <math>\sin x + \cos x = \sqrt{\frac{n}{10}}</math>. Squaring both sides gives us
 +
<cmath>\sin^2 x + 2\sin x \cos x + \cos^2 x = \frac{n}{10}</cmath>
 +
 +
Via the Pythagorean Identity, <math>\sin^2 x + \cos^2 x = 1</math> and <math>2\sin x \cos x</math> is simply <math>\frac{1}{5}</math>, via substitution. Thus, substituting these results into the current equation:
 +
<cmath>1 + \frac{1}{5} = \frac{n}{10}</cmath>
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<cmath>\implies \frac{6}{5} = \frac{n}{10}</cmath>
 +
 +
Using simple cross-multiplication techniques, we have <math>5n = 60</math>, and thus <math>\boxed{n = 012}</math>.
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~ nikenissan
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 +
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==Solution 3==
 +
By the first equation, we get that <math>\sin(x)*\cos(x)=10^{-1}</math>. We can let <math>\sin(x)=a</math>, <math>\cos(x)=b</math>. Thus <math>ab=\frac{1}{10}</math>. By the identity <math>\sin^2x+\cos^2x=1</math>, we get that <math>a^2+b^2=1</math>. Solving this, we get <math>a+b=\sqrt{\frac{12}{10}}</math>. So we have
 +
 +
<cmath>\log\left(\sqrt{\frac{12}{10}}\right)=\frac12(\log(n)-1)</cmath>
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<cmath>2\log\left(\sqrt{\frac{12}{10}}\right)=\log(n)-1</cmath>
 +
<cmath>\log\left(\frac{12}{10}\right)+1=\log(n)</cmath>
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<cmath>\log\left(\frac{12}{10}\right)+\log(10)=\log(n)</cmath>
 +
<cmath>\log\left(\frac{12}{10}\times 10\right)=\log(12)=\log(n)</cmath>
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 +
From here it is obvious that <math>\boxed{n=012}</math>.
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 +
 +
~yofro
 +
 +
==Solution 4==
 +
Let <math>\log{x} = \log_{10}{x}.</math> Through basic log properties, we see that <math>\log{a} + \log{b} = \log{(ab)}.</math> Thus, we see that <math>\log{(\sin{x})} + \log{(\cos{x})} = \log{(\sin{x}\cos{x})} = -1.</math> Simplifying, we get:
 +
 +
\begin{align*}
 +
\log{(\sin{x}\cos{x})} &= -1 \\
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\sin{x}\cos{x} &= 10^{-1} = \frac{1}{10}
 +
\end{align*}
 +
 +
Next, we can manipulate the second equation to get:
 +
 +
\begin{align*}
 +
\log{(\sin{x} + \cos{x})} &= \frac{1}{2}(\log{n}-1) \\
 +
2\log{(\sin{x} + \cos{x})} &= \log{n}-1 \\
 +
\log{(\sin{x} + \cos{x})^2} + 1 &= \log{n}
 +
\end{align*}
 +
 +
Expanding <math>(\sin{x} + \cos{x})^2,</math> we get:
 +
 +
\begin{align*}
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\log{(\sin^2{x} + \cos^2{x} + 2\sin{x}\cos{x})} + 1 &= \log{n} \\
 +
\log{(1 + 2\sin{x}\cos{x})} + 1 &= \log{n} \\
 +
\log{(1 + \frac{2}{10})} + \log{10} &= \log{n} \\
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\log{(\frac{12}{10} \cdot 10)} = \log{n} \\
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\log{12} = \log{n}
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\end{align*}
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Finally, we see that <math>n = \boxed{012}.</math>
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 +
~ Cheetahboy93
  
 
== See also ==
 
== See also ==

Latest revision as of 14:30, 22 September 2024

Problem

Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1),$ find $n.$

Solution 1

Using the properties of logarithms, we can simplify the first equation to $\log_{10} \sin x + \log_{10} \cos x = \log_{10}(\sin x \cos x) = -1$. Therefore, \[\sin x \cos x = \frac{1}{10}.\qquad (*)\]

Now, manipulate the second equation. \begin{align*} \log_{10} (\sin x + \cos x) &= \frac{1}{2}(\log_{10} n - \log_{10} 10) \\ \log_{10} (\sin x + \cos x) &= \left(\log_{10} \sqrt{\frac{n}{10}}\right) \\ \sin x + \cos x &= \sqrt{\frac{n}{10}} \\ (\sin x + \cos x)^{2} &= \left(\sqrt{\frac{n}{10}}\right)^2 \\ \sin^2 x + \cos^2 x +2 \sin x \cos x &= \frac{n}{10} \\ \end{align*}

By the Pythagorean identities, $\sin ^2 x + \cos ^2 x = 1$, and we can substitute the value for $\sin x \cos x$ from $(*)$. $1 + 2\left(\frac{1}{10}\right) = \frac{n}{10} \Longrightarrow n = \boxed{012}$.

Solution 2

Examining the first equation, we simplify as the following: \[\log_{10} \sin x \cos x = -1\] \[\implies \sin x \cos x = \frac{1}{10}\]

With this in mind, examining the second equation, we may simplify as the following (utilizing logarithm properties): \[\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - \log_{10} 10)\] \[\implies \log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} \frac{n}{10})\] \[\implies \log_{10} (\sin x + \cos x) = \log_{10} \sqrt{\frac{n}{10}}\]

From here, we may divide both sides by $\log_{10} (\sin x + \cos x)$ and then proceed with the change-of-base logarithm property: \[1 = \frac{\log_{10} \sqrt{\frac{n}{10}}}{\log_{10} (\sin x + \cos x)}\] \[\implies 1 = \log_{\sin x + \cos x} \sqrt{\frac{n}{10}}\]

Thus, exponentiating both sides results in $\sin x + \cos x = \sqrt{\frac{n}{10}}$. Squaring both sides gives us \[\sin^2 x + 2\sin x \cos x + \cos^2 x = \frac{n}{10}\]

Via the Pythagorean Identity, $\sin^2 x + \cos^2 x = 1$ and $2\sin x \cos x$ is simply $\frac{1}{5}$, via substitution. Thus, substituting these results into the current equation: \[1 + \frac{1}{5} = \frac{n}{10}\] \[\implies \frac{6}{5} = \frac{n}{10}\]

Using simple cross-multiplication techniques, we have $5n = 60$, and thus $\boxed{n = 012}$. ~ nikenissan


Solution 3

By the first equation, we get that $\sin(x)*\cos(x)=10^{-1}$. We can let $\sin(x)=a$, $\cos(x)=b$. Thus $ab=\frac{1}{10}$. By the identity $\sin^2x+\cos^2x=1$, we get that $a^2+b^2=1$. Solving this, we get $a+b=\sqrt{\frac{12}{10}}$. So we have

\[\log\left(\sqrt{\frac{12}{10}}\right)=\frac12(\log(n)-1)\] \[2\log\left(\sqrt{\frac{12}{10}}\right)=\log(n)-1\] \[\log\left(\frac{12}{10}\right)+1=\log(n)\] \[\log\left(\frac{12}{10}\right)+\log(10)=\log(n)\] \[\log\left(\frac{12}{10}\times 10\right)=\log(12)=\log(n)\]

From here it is obvious that $\boxed{n=012}$.


~yofro

Solution 4

Let $\log{x} = \log_{10}{x}.$ Through basic log properties, we see that $\log{a} + \log{b} = \log{(ab)}.$ Thus, we see that $\log{(\sin{x})} + \log{(\cos{x})} = \log{(\sin{x}\cos{x})} = -1.$ Simplifying, we get:

\begin{align*} \log{(\sin{x}\cos{x})} &= -1 \\ \sin{x}\cos{x} &= 10^{-1} = \frac{1}{10} \end{align*}

Next, we can manipulate the second equation to get:

\begin{align*} \log{(\sin{x} + \cos{x})} &= \frac{1}{2}(\log{n}-1) \\ 2\log{(\sin{x} + \cos{x})} &= \log{n}-1 \\ \log{(\sin{x} + \cos{x})^2} + 1 &= \log{n} \end{align*}

Expanding $(\sin{x} + \cos{x})^2,$ we get:

\begin{align*} \log{(\sin^2{x} + \cos^2{x} + 2\sin{x}\cos{x})} + 1 &= \log{n} \\ \log{(1 + 2\sin{x}\cos{x})} + 1 &= \log{n} \\ \log{(1 + \frac{2}{10})} + \log{10} &= \log{n} \\ \log{(\frac{12}{10} \cdot 10)} = \log{n} \\ \log{12} = \log{n} \end{align*}

Finally, we see that $n = \boxed{012}.$

~ Cheetahboy93

See also

2003 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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All AIME Problems and Solutions

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