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?</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><math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math>
  
==Solution==
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==Solution 1==
{{solution}}
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First, move terms to get <math>1+5\cos 3x=3\sin x</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5\cos 3x</math>). -dstanz5
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==Solution 2==
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We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath>
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Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with
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amplitude 5 and period <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sin function
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with amplitude 3 and shifted 1 unit downwards. So:
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<asy>
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import graph;
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size(400,200,IgnoreAspect);
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real Sin(real t) {return 3*sin(t) - 1;}
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real Cos(real t) {return 5*cos(3*t);}
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draw(graph(Sin,0, 2pi),red,"$3\sin{x} -1 $");
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draw(graph(Cos,0, 2pi),blue,"$5\cos{3x}$");
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xaxis("$x$",BottomTop,LeftTicks);
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yaxis("$y$",LeftRight,RightTicks(trailingzero));
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add(legend(),point(E),20E,UnFill);
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</asy>
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We have <math>\boxed{(A) 6}</math> solutions. ~Jamess2022 (burntTacos)
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== Video Solution by OmegaLearn (Using Sine and Cosine Graph) ==
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https://youtu.be/toBOpc6vS6s
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~ 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}}

Revision as of 15:57, 5 March 2021

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 1

First, move terms to get $1+5\cos 3x=3\sin x$. After graphing, we find that there are $\boxed{6}$ solutions (two in each period of $5\cos 3x$). -dstanz5


Solution 2

We can graph two functions in this case: $5\cos{3x}$ and $3\sin{x} -1$. \[\newline\] Using transformation of functions, we know that $5\cos{3x}$ is just a cos function with amplitude 5 and period $\frac{2\pi}{3}$. Similarly, $3\sin{x} -1$ is just a sin function with amplitude 3 and shifted 1 unit downwards. So: [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] We have $\boxed{(A) 6}$ solutions. ~Jamess2022 (burntTacos)

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
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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