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 1== |
First, move terms to get <math>1+5cos3x=3sinx</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5cos3x</math>). -dstanz5 | First, move terms to get <math>1+5cos3x=3sinx</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5cos3x</math>). -dstanz5 | ||
| − | ==Solution | + | ==Solution 2== |
We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath> | We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath> | ||
Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with | Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with | ||
Revision as of 12:10, 12 February 2021
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 1
First, move terms to get 
. After graphing, we find that there are 
solutions (two in each period of 
). -dstanz5
Solution 2
We can graph two functions in this case: 
and 
. ![\[\newline\]](http://latex.artofproblemsolving.com/c/7/2/c721557bcaa86e38c25ffa8c48f3b057b9f631e2.png)
Using transformation of functions, we know that is just a cos function with
amplitude 5 and frequency 
. Similarly, 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]](http://latex.artofproblemsolving.com/3/d/9/3d9389c15400dfa88ba64f7bb7be2879cf83e6c4.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. 
