Difference between revisions of "2003 AIME I Problems/Problem 4"

Revision as of 19:51, 13 April 2017

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

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 $(*)$ . $1 + 2\left(\frac{1}{10}\right) = \frac{n}{10} \Longrightarrow n = \boxed{012}$ .

@@ Line 1: / Line 1: @@
 == 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 - 10), </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>
 == Solution ==

2003 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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Difference between revisions of "2003 AIME I Problems/Problem 4"

Revision as of 19:51, 13 April 2017

Problem

Solution

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