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

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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>
 
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 2==
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==Solution==
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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We rearrange to get <cmath>5\cos(3\theta) = 3\sin(\theta) - 1.</cmath>
Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with
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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>.
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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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 downwards. So:
with amplitude 3 and shifted 1 unit downwards. So:
 
 
<asy>
 
<asy>
 
import graph;
 
import graph;
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add(legend(),point(E),20E,UnFill);
 
add(legend(),point(E),20E,UnFill);
 
</asy>
 
</asy>
We have <math>\boxed{\textbf{(D) }6}</math> solutions. ~Jamess2022 (burntTacos)
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We have <math>\boxed{\textbf{(D) }6}</math> solutions.  
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~Jamess2022 (burntTacos)
  
 
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) ==
 
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) ==

Revision as of 03:15, 28 January 2022

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 \[5\cos(3\theta) = 3\sin(\theta) - 1.\] 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 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 $\boxed{\textbf{(D) }6}$ 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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