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 | + | ==Solution== |
| − | We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. | + | 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 | + | 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 | + | 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) | + | We have <math>\boxed{\textbf{(D) }6}</math> solutions. |
| + | |||
| + | ~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
Contents
Problem
How many values of 
in the interval 
satisfy![\[1-3\sin\theta+5\cos3\theta = 0?\]](http://latex.artofproblemsolving.com/b/5/8/b58210a8828d746350c9d6235739f48869ca7cf6.png)
Solution
We rearrange to get ![\[5\cos(3\theta) = 3\sin(\theta) - 1.\]](http://latex.artofproblemsolving.com/4/9/9/499a596f613ba3727892091a3fb415de0940db3a.png)
We can graph two functions in this case: 
and 
.
Using transformation of functions, we know that 
is just a cosine function with amplitude 
and period 
. Similarly, 
is just a sine function with amplitude 
and shifted 
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]](http://latex.artofproblemsolving.com/a/c/0/ac0756ac2aff279bb152a3f7a2f32a2fc7319a66.png)
We have 
solutions.
~Jamess2022 (burntTacos)
Video Solution by OmegaLearn (Using Sine and Cosine Graph)
~ 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
