Difference between revisions of "2021 AMC 12B Problems/Problem 13"

Latest revision as of 17:57, 19 September 2023

Problem

How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy $1-3\sin\theta+5\cos3\theta = 0?$ $\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$

Solution

We rearrange to get $5\cos3\theta = 3\sin\theta-1.$ We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$ . Using transformation of functions, we know that $5\cos{3x}$ is just a cosine function with amplitude $5$ and period $\frac{2\pi}{3}$ . Similarly, $3\sin{x} -1$ is just a sine function with amplitude $3$ and shifted $1$ 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 $\boxed{\textbf{(D) }6}$ solutions.

~Jamess2022 (burntTacos)

Video Solution (Just 1 min!)

https://youtu.be/2wYcntg1mCc

~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)

https://youtu.be/toBOpc6vS6s

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

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.

@@ Line 1: / Line 1: @@
 ==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?</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;
-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>)
+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) ==
+https://youtu.be/toBOpc6vS6s
+~ pi_is_3.14
+==Video Solution by Hawk Math==
+https://www.youtube.com/watch?v=p4iCAZRUESs
 ==See Also==
 {{AMC12 box|year=2021|ab=B|num-b=12|num-a=14}}
 {{MAA Notice}}

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