Difference between revisions of "2021 AMC 12B Problems/Problem 13"

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==Problem==
 
==Problem==
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath><math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math>
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How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy <cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath>
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<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math>
  
 
==Solution==
 
==Solution==
 
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We rearrange to get <cmath>5\cos3\theta = 3\sin\theta-1.</cmath>
First, move terms to get <math>1+5cos3x=3sinx</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5cos3x</math>). -dstanz5
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We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </math>.
 
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Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cosine function with amplitude <math>5</math> and period <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sine function with amplitude <math>3</math> and shifted <math>1</math> unit downward:
 
 
==Solution 1==
 
We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath>
 
Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with
 
amplitude 5 and frequency <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sin function
 
with amplitude 3 and shifted 1 unit downwards. So:
 
 
<asy>
 
<asy>
 
import graph;
 
import graph;
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yaxis("$y$",LeftRight,RightTicks(trailingzero));
 
yaxis("$y$",LeftRight,RightTicks(trailingzero));
  
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add(legend(),point(E),20E,UnFill);
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</asy>
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So, we have <math>\boxed{\textbf{(D) }6}</math> solutions.
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~Jamess2022 (burntTacos)
  
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==Video Solution (Just 1 min!)==
 +
https://youtu.be/2wYcntg1mCc
  
add(legend(),point(E),20E,UnFill);
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<i>~Education, the Study of Everything </i>
</asy>
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We have <math>\boxed{(A) 6}</math> solutions. ~Jamess2022 (burntTacos)
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==Video Solution (quick, no graphing)==
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https://youtu.be/YTn5YPQt6IY
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~ MathProblemSolvingSkills.com
  
 
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) ==
 
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) ==
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~ pi_is_3.14
 
~ pi_is_3.14
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==Video Solution by Hawk Math==
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https://www.youtube.com/watch?v=p4iCAZRUESs
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2021|ab=B|num-b=12|num-a=14}}
 
{{AMC12 box|year=2021|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:57, 19 September 2023

Problem

How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\] $\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$

Solution

We rearrange to get \[5\cos3\theta = 3\sin\theta-1.\] We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$. Using transformation of functions, we know that $5\cos{3x}$ is just a cosine function with amplitude $5$ and period $\frac{2\pi}{3}$. Similarly, $3\sin{x} -1$ is just a sine function with amplitude $3$ and shifted $1$ unit downward: [asy] import graph;  size(400,200,IgnoreAspect);  real Sin(real t) {return 3*sin(t) - 1;} real Cos(real t) {return 5*cos(3*t);}  draw(graph(Sin,0, 2pi),red,"$3\sin{x} -1 $"); draw(graph(Cos,0, 2pi),blue,"$5\cos{3x}$");  xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks(trailingzero));  add(legend(),point(E),20E,UnFill); [/asy] So, we have $\boxed{\textbf{(D) }6}$ solutions.

~Jamess2022 (burntTacos)

Video Solution (Just 1 min!)

https://youtu.be/2wYcntg1mCc

~Education, the Study of Everything

Video Solution (quick, no graphing)

https://youtu.be/YTn5YPQt6IY ~ MathProblemSolvingSkills.com

Video Solution by OmegaLearn (Using Sine and Cosine Graph)

https://youtu.be/toBOpc6vS6s

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

See Also

2021 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AMC 12 Problems and Solutions

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