# 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

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)

~ pi_is_3.14

## Video Solution by Hawk Math

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