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 $1+5cos3x=3sinx$ . After graphing, we find that there are $\boxed{6}$ solutions (two in each period of $5cos3x$ ). -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

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

Page

Toolbox

Search