2021 AMC 12B Problems/Problem 13

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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

See Also

2021 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AMC 12 Problems and Solutions

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