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Difference between revisions of "2021 Fall AMC 12A Problems/Problem 19"

(→‎See Also)
(→‎Video Solution by Mathematical Dexterity)
 
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==Problem 19==
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==Problem==
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Let <math>x</math> be the least real number greater than <math>1</math> such that sin<math>(x)</math> = sin<math>(x^2)</math>, where the arguments are in degrees. What is <math>x</math> rounded up to the closest integer?
+
Let <math>x</math> be the least real number greater than <math>1</math> such that <math>\sin(x)= \sin(x^2)</math>, where the arguments are in degrees. What is <math>x</math> rounded up to the closest integer?
  
 
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20</math>
 
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20</math>
Line 14: Line 14:
  
 
~kingofpineapplz
 
~kingofpineapplz
 +
 +
== Solution 2 ==
 +
For choice <math>\textbf{(A)},</math> we have
 +
<cmath>
 +
\begin{align*}
 +
\left| \sin x - \sin \left( x^2 \right) \right|
 +
& = \left| \sin 10^\circ - \sin \left( \left( 10^2 \right)^\circ \right) \right| \\
 +
& = \left| \sin 10^\circ - \sin 100^\circ \right| \\
 +
& = \left| \sin 10^\circ - \sin 80^\circ \right| \\
 +
& = \sin 80^\circ - \sin 10^\circ .
 +
\end{align*}
 +
</cmath>
 +
For choice <math>\textbf{(B)},</math> we have
 +
<cmath>
 +
\begin{align*}
 +
\left| \sin x - \sin \left( x^2 \right) \right|
 +
& = \left| \sin 13^\circ - \sin \left( \left( 13^2 \right)^\circ \right) \right| \\
 +
& = \left| \sin 13^\circ - \sin 169^\circ \right| \\
 +
& = \left| \sin 10^\circ - \sin 11^\circ \right| \\
 +
& = \sin 11^\circ - \sin 10^\circ .
 +
\end{align*}
 +
</cmath>
 +
For choice <math>\textbf{(C)},</math> we have
 +
<cmath>
 +
\begin{align*}
 +
\left| \sin x - \sin \left( x^2 \right) \right|
 +
& = \left| \sin 14^\circ - \sin \left( \left( 14^2 \right)^\circ \right) \right| \\
 +
& = \left| \sin 14^\circ - \sin 196^\circ \right| \\
 +
& = \left| \sin 14^\circ + \sin 16^\circ \right| \\
 +
& = \sin 14^\circ + \sin 16^\circ .
 +
\end{align*}
 +
</cmath>
 +
For choice <math>\textbf{(D)},</math> we have
 +
<cmath>
 +
\begin{align*}
 +
\left| \sin x - \sin \left( x^2 \right) \right|
 +
& = \left| \sin 19^\circ - \sin \left( \left( 19^2 \right)^\circ \right) \right| \\
 +
& = \left| \sin 19^\circ - \sin 361^\circ \right| \\
 +
& = \left| \sin 19^\circ - \sin 1^\circ \right| \\
 +
& = \sin 19^\circ - \sin 1^\circ .
 +
\end{align*}
 +
</cmath>
 +
For choice <math>\textbf{(E)},</math> we have
 +
<cmath>
 +
\begin{align*}
 +
\left| \sin x - \sin \left( x^2 \right) \right|
 +
& = \left| \sin 20^\circ - \sin \left( \left( 20^2 \right)^\circ \right) \right| \\
 +
& = \left| \sin 20^\circ - \sin 400^\circ \right| \\
 +
& = \left| \sin 20^\circ - \sin 40^\circ \right| \\
 +
& = \sin 40^\circ - \sin 20^\circ .
 +
\end{align*}
 +
</cmath>
 +
Therefore, the answer is <math>\boxed{\textbf{(B) }13},</math> as <math>\sin 11^\circ - \sin 10^\circ</math> is the closest to <math>0.</math>
 +
 +
~Steven Chen (www.professorchenedu.com)
 +
 +
==Easy Solution with Intermediate Value Theorem==
 +
we know that x^2-x=180-2x+360k or x^2-x=360k from looking at the sin values on the unit circle.
 +
with these equations, we can just insert the values that are given in the 5 answers.
 +
let's try A first.
 +
we know that one of the equations should be greater on the left, and another equation we make should be greater on the right for the solution to be in between the values inserted into the equations
 +
the two numbers it is in between is 9 and 10 since it is rounded up.
 +
100-10 vs 180-20
 +
81-9 vs 180-18
 +
this does not satisfy what we would like to find
 +
now let's try B
 +
between 12 and 13
 +
144-12 vs 180-24
 +
169-13 vs 180-26
 +
now this satisfies our hunt for the solution
 +
the answer is B
 +
 +
emilyyunhanq@gmail.com
 +
solution by Emily Q
 +
 +
==Video Solution by Mathematical Dexterity==
 +
https://www.youtube.com/watch?v=H0pNJFbV4jE
 +
 +
==Video Solution by TheBeautyofMath==
 +
Solved both Mentally and by writing things down
 +
 +
https://youtu.be/o2MAmtgBbKc
 +
 +
~IceMatrix
  
 
==See Also==
 
==See Also==
−
{{AMC12 box|year=2021 Fall|ab=A|before=First Problem|num-a=19}}
+
{{AMC12 box|year=2021 Fall|ab=A|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 16:27, 18 May 2024

Problem

Let $x$ be the least real number greater than $1$ such that $\sin(x)= \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer?

$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20$

Solution 1

The smallest $x$ to make $\sin(x) = \sin(x^2)$ would require $x=x^2$, but since $x$ needs to be greater than $1$, these solutions are not valid.

The next smallest $x$ would require $x=180-x^2$, or $x^2+x=180$.

After a bit of guessing and checking, we find that $12^2+12=156$, and $13^2+13=182$, so the solution lies between $12{ }$ and $13$, making our answer $\boxed{\textbf{(B) } 13}.$

Note: One can also solve the quadratic and estimate the radical.

~kingofpineapplz

Solution 2

For choice $\textbf{(A)},$ we have \begin{align*} \left| \sin x - \sin \left( x^2 \right) \right| & = \left| \sin 10^\circ - \sin \left( \left( 10^2 \right)^\circ \right) \right| \\ & = \left| \sin 10^\circ - \sin 100^\circ \right| \\ & = \left| \sin 10^\circ - \sin 80^\circ \right| \\ & = \sin 80^\circ - \sin 10^\circ . \end{align*} For choice $\textbf{(B)},$ we have \begin{align*} \left| \sin x - \sin \left( x^2 \right) \right| & = \left| \sin 13^\circ - \sin \left( \left( 13^2 \right)^\circ \right) \right| \\ & = \left| \sin 13^\circ - \sin 169^\circ \right| \\ & = \left| \sin 10^\circ - \sin 11^\circ \right| \\ & = \sin 11^\circ - \sin 10^\circ . \end{align*} For choice $\textbf{(C)},$ we have \begin{align*} \left| \sin x - \sin \left( x^2 \right) \right| & = \left| \sin 14^\circ - \sin \left( \left( 14^2 \right)^\circ \right) \right| \\ & = \left| \sin 14^\circ - \sin 196^\circ \right| \\ & = \left| \sin 14^\circ + \sin 16^\circ \right| \\ & = \sin 14^\circ + \sin 16^\circ . \end{align*} For choice $\textbf{(D)},$ we have \begin{align*} \left| \sin x - \sin \left( x^2 \right) \right| & = \left| \sin 19^\circ - \sin \left( \left( 19^2 \right)^\circ \right) \right| \\ & = \left| \sin 19^\circ - \sin 361^\circ \right| \\ & = \left| \sin 19^\circ - \sin 1^\circ \right| \\ & = \sin 19^\circ - \sin 1^\circ . \end{align*} For choice $\textbf{(E)},$ we have \begin{align*} \left| \sin x - \sin \left( x^2 \right) \right| & = \left| \sin 20^\circ - \sin \left( \left( 20^2 \right)^\circ \right) \right| \\ & = \left| \sin 20^\circ - \sin 400^\circ \right| \\ & = \left| \sin 20^\circ - \sin 40^\circ \right| \\ & = \sin 40^\circ - \sin 20^\circ . \end{align*} Therefore, the answer is $\boxed{\textbf{(B) }13},$ as $\sin 11^\circ - \sin 10^\circ$ is the closest to $0.$

~Steven Chen (www.professorchenedu.com)

Easy Solution with Intermediate Value Theorem

we know that x^2-x=180-2x+360k or x^2-x=360k from looking at the sin values on the unit circle. with these equations, we can just insert the values that are given in the 5 answers. let's try A first. we know that one of the equations should be greater on the left, and another equation we make should be greater on the right for the solution to be in between the values inserted into the equations the two numbers it is in between is 9 and 10 since it is rounded up. 100-10 vs 180-20 81-9 vs 180-18 this does not satisfy what we would like to find now let's try B between 12 and 13 144-12 vs 180-24 169-13 vs 180-26 now this satisfies our hunt for the solution the answer is B

emilyyunhanq@gmail.com solution by Emily Q

Video Solution by Mathematical Dexterity

https://www.youtube.com/watch?v=H0pNJFbV4jE

Video Solution by TheBeautyofMath

Solved both Mentally and by writing things down

https://youtu.be/o2MAmtgBbKc

~IceMatrix

See Also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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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