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Difference between revisions of "2010 AMC 12B Problems/Problem 13"

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We start by noticing that the maximum values for both the sine and the cosine function are 1.  
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== Problem ==
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Therefore, the only way for this equation to be true is if <math>\cos(2A-B)=1</math> and <math>\sin(A+B)=1</math>, since if one of these equaled less than 1, the other one would have to be greater than 1 which contradicts our previous statement.
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In <math>\triangle ABC</math>, <math>\cos(2A-B)+\sin(A+B)=2</math> and <math>AB=4</math>. What is <math>BC</math>?
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From this we easily conclude that <math>2A-B=0</math> and <math>A+B=90</math>. Solving this system gives us <math>A=30</math> and <math>B=60</math>, which gives us a <math>30-60-90</math> triangle. By drawing a diagram we can easily see that <math>BC=2</math>　<math>(C)</math>
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<math>\textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}</math>
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== Solution ==
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We note that <math>-1</math> <math>\le</math> <math>\sin x</math> <math>\le</math> <math>1</math> and <math>-1</math> <math>\le</math> <math>\cos x</math> <math>\le</math> <math>1</math>.  
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Therefore, there is no other way to satisfy this equation other than making both <math>\cos(2A-B)=1</math> and <math>\sin(A+B)=1</math>, since any other way would cause one of these values to become greater than 1, which contradicts our previous statement.
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From this we can easily conclude that <math>2A-B=0^{\circ}</math> and <math>A+B=90^{\circ}</math> and solving this system gives us <math>A=30^{\circ}</math> and <math>B=60^{\circ}</math>. It is clear that <math>\triangle ABC</math> is a <math>30^{\circ},60^{\circ},90^{\circ}</math> triangle with <math>BC=2</math>　<math>\Longrightarrow</math>　<math>(C)</math>.
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== See also ==
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{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}
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[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 23:31, 17 January 2020

Problem

In $\triangle ABC$, $\cos(2A-B)+\sin(A+B)=2$ and $AB=4$. What is $BC$?

$\textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$

Solution

We note that $-1$ $\le$ $\sin x$ $\le$ $1$ and $-1$ $\le$ $\cos x$ $\le$ $1$. Therefore, there is no other way to satisfy this equation other than making both $\cos(2A-B)=1$ and $\sin(A+B)=1$, since any other way would cause one of these values to become greater than 1, which contradicts our previous statement. From this we can easily conclude that $2A-B=0^{\circ}$ and $A+B=90^{\circ}$ and solving this system gives us $A=30^{\circ}$ and $B=60^{\circ}$. It is clear that $\triangle ABC$ is a $30^{\circ},60^{\circ},90^{\circ}$ triangle with $BC=2$　$\Longrightarrow$　$(C)$.

See also

2010 AMC 12B (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
All AMC 12 Problems and Solutions

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