Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2020 AMC 12A Problems/Problem 9

Revision as of 13:15, 1 February 2020 by Lopkiloinm (talk | contribs)

Problem

How many solutions does the equation tan$(2x)=cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Draw a graph of tan$(2x)$ and $cos(\frac{x}{2})$

tan$(2x)$ has a period of $\frac{\pi}{2}$ and asymptotes at $\frac{\pi}{4}+\frac{k\pi}{2}$.

$cos(\frac{x}{2})$ has a period of $4\pi$ and zeroes at $\pi$.

Drawing such a graph would get $\boxed{\textbf{E) }5}$ ~lopkiloinm

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AMC_12A_Problems/Problem_9&oldid=116283"