Difference between revisions of "2021 AMC 12B Problems/Problem 13"
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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 1== | ||
| + | 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 | ||
| + | amplitude 5 and frequency <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> 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> | ||
| + | We have <math>\boxed{(A) 6}</math> solutions. | ||
== Video Solution by OmegaLearn (Using Sine and Cosine Graph) == | == Video Solution by OmegaLearn (Using Sine and Cosine Graph) == | ||
Revision as of 06:19, 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
First, move terms to get 
. After graphing, we find that there are 
solutions (two in each period of 
). -dstanz5
Solution 1
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.
Video Solution by OmegaLearn (Using Sine and Cosine Graph)
~ pi_is_3.14
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 | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
