Difference between revisions of "2020 AMC 12A Problems/Problem 9"
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Draw a graph of <math>\tan(2x)</math> and <math>\cos(\tfrac{x}{2})</math> | Draw a graph of <math>\tan(2x)</math> and <math>\cos(\tfrac{x}{2})</math> | ||
| − | <math>\tan(2x)</math> has a period of <math>\ | + | <math>\tan(2x)</math> has a period of <math>\tfrac{\pi}{2},</math> asymptotes at <math>x = \tfrac{\pi}{4}+\tfrac{k\pi}{2},</math> and zeroes at <math>\tfrac{k\pi}{2}</math>. It is positive from <math>(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})</math> and negative elsewhere. |
cos<math>(\tfrac{x}{2})</math> has a period of <math>4\pi</math> and zeroes at <math>\pi</math>. It is positive from <math>[0,\pi)</math> and negative elsewhere. | cos<math>(\tfrac{x}{2})</math> has a period of <math>4\pi</math> and zeroes at <math>\pi</math>. It is positive from <math>[0,\pi)</math> and negative elsewhere. | ||
Revision as of 20:59, 7 March 2020
Problem
How many solutions does the equation 
have on the interval ![$[0,2\pi]?$](http://latex.artofproblemsolving.com/e/c/c/ecccefb4c460857891d99273237d9deb9ecf41c1.png)
Solution
Draw a graph of 
and 
has a period of 
asymptotes at 
and zeroes at 
. It is positive from 
and negative elsewhere.
cos
has a period of 
and zeroes at 
. It is positive from 
and negative elsewhere.
Drawing such a graph would get 
~lopkiloinm
edited by - annabelle0913
See Also
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
