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> | + | 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 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>. | ||
| + | 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> | ||
| + | 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 <math>\boxed{\textbf{(D) }6}</math> solutions. | ||
| + | |||
| + | ~Jamess2022 (burntTacos) | ||
| + | |||
| + | ==Video Solution (Just 1 min!)== | ||
| + | https://youtu.be/2wYcntg1mCc | ||
| + | |||
| + | <i>~Education, the Study of Everything </i> | ||
| + | |||
| + | ==Video Solution (quick, no graphing)== | ||
| + | https://youtu.be/YTn5YPQt6IY | ||
| + | ~ MathProblemSolvingSkills.com | ||
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) == | == Video Solution by OmegaLearn (Using Sine and Cosine Graph) == | ||
| Line 11: | Line 40: | ||
~ pi_is_3.14 | ~ pi_is_3.14 | ||
| + | |||
| + | ==Video Solution by Hawk Math== | ||
| + | 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 18:57, 19 September 2023
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 ![\[5\cos3\theta = 3\sin\theta-1.\]](http://latex.artofproblemsolving.com/6/4/0/6403161c0a12179aebce70cb4619ca393d783ee5.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 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]](http://latex.artofproblemsolving.com/a/c/0/ac0756ac2aff279bb152a3f7a2f32a2fc7319a66.png)
So, we have 
solutions.
~Jamess2022 (burntTacos)
Video Solution (Just 1 min!)
~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)
~ 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. 
