Difference between revisions of "2021 AMC 12B Problems/Problem 13"
MRENTHUSIASM (talk | contribs) m |
MRENTHUSIASM (talk | contribs) m (→Solution) |
||
| Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
| − | We rearrange to get <cmath>5\ | + | 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. So: | 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: | ||
Revision as of 03:16, 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 ![\[5\cos3\theta = 3\sin\theta - 1.\]](http://latex.artofproblemsolving.com/7/c/6/7c6ceb4aca10addbeeebb3d631cd887045be73ec.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. 
