2020 AMC 12A Problems/Problem 9: Difference between revisions

Revision as of 13:15, 1 February 2020

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

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

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

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 <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>
+==Solution==
+Draw a graph of tan<math>(2x)</math> and <math>cos(\frac{x}{2})</math>
+tan<math>(2x)</math> has a period of <math>\frac{\pi}{2}</math> and asymptotes at <math>\frac{\pi}{4}+\frac{k\pi}{2}</math>.
+<math>cos(\frac{x}{2})</math> has a period of <math>4\pi</math> and zeroes at <math>\pi</math>.
+Drawing such a graph would get <math>\boxed{\textbf{E) }5}</math> ~lopkiloinm