# 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)

~ pi_is_3.14

## See Also

 2021 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 12 Followed byProblem 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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