Difference between revisions of "2020 AMC 12A Problems/Problem 9"

Revision as of 14:59, 17 May 2020

Problem

How many solutions does the equation $\tan(2x)=\cos(\tfrac{x}{2})$ 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 $\cos(\tfrac{x}{2})$

$\tan(2x)$ has a period of $\tfrac{\pi}{2},$ asymptotes at $x = \tfrac{\pi}{4}+\tfrac{k\pi}{2},$ and zeroes at $\tfrac{k\pi}{2}$ . It is positive from $(0,\tfrac{\pi}{4}) \cup (\tfrac{\pi}{2},\tfrac{3\pi}{4}) \cup (\pi,\tfrac{5\pi}{4}) \cup (\tfrac{3\pi}{2},\tfrac{7\pi}{4})$ and negative elsewhere.

cos $(\tfrac{x}{2})$ has a period of $4\pi$ and zeroes at $\pi$ . It is positive from $[0,\pi)$ and negative elsewhere.

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

Or you could see the points at which both graphs are positive or both are negative, again yielding 5 such areas. -hi13

edited by - annabelle0913

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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.

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Difference between revisions of "2020 AMC 12A Problems/Problem 9"

Revision as of 14:59, 17 May 2020

Problem

Solution

See Also

@@ Line 14: / Line 14: @@
 Drawing such a graph would get <math>\boxed{\textbf{E) }5}</math> ~lopkiloinm
+Or you could see the points at which both graphs are positive or both are negative, again yielding 5 such areas. -hi13
 edited by - annabelle0913