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

m (→‎Solution)
m (→‎Solution)
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We rearrange to get <cmath>5\cos3\theta = 3\sin\theta-1.</cmath>
 
We rearrange to get <cmath>5\cos3\theta = 3\sin\theta-1.</cmath>
 
We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </math>.
 
We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </math>.
−
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:
+
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:
 
<asy>
 
<asy>
 
import graph;
 
import graph;
Line 22: Line 22:
 
add(legend(),point(E),20E,UnFill);
 
add(legend(),point(E),20E,UnFill);
 
</asy>
 
</asy>
−
We have <math>\boxed{\textbf{(D) }6}</math> solutions.  
+
So, we have <math>\boxed{\textbf{(D) }6}</math> solutions.  
  
 
~Jamess2022 (burntTacos)
 
~Jamess2022 (burntTacos)

Revision as of 03:21, 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 \[5\cos3\theta = 3\sin\theta-1.\] 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 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 (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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