2021 AMC 12B Problems/Problem 13

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 1

First, move terms to get $1+5\cos 3x=3\sin x$. After graphing, we find that there are $\boxed{6}$ solutions (two in each period of $5\cos 3x$). -dstanz5


Solution 2

We can graph two functions in this case: $5\cos{3x}$ and $3\sin{x} -1$. \[\newline\] Using transformation of functions, we know that $5\cos{3x}$ is just a cos function with amplitude 5 and period $\frac{2\pi}{3}$. Similarly, $3\sin{x} -1$ 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 $\boxed{(A) 6}$ solutions. ~Jamess2022 (burntTacos)

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
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

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