2021 AMC 12A Problems/Problem 19
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Contents
Problem
How many solutions does the equation 
have in the closed interval ![$[0,\pi]$](http://latex.artofproblemsolving.com/e/c/e/ece27ba559bd93cdf2416aa0f88fc8d7d85e5791.png)
?
Solution 1 (Inverse Trigonometric Functions)
The ranges of 
and 
are both ![$\left[-\frac{\pi}2, \frac{\pi}2 \right]$](http://latex.artofproblemsolving.com/b/0/b/b0bb335ab4991f0acbd8f690519b105d93676bf4.png)
, which is included in the range of 
, so we can use it with no issues.
This only happens at 
on the interval , because one of 
and 
must be 
and the other 
. Therefore, the answer is 
~Tucker
Solution 2 (Analysis)
Let 
and 
This problem is equivalent to counting the intersections of the graphs of 
and 
in the closed interval ![$[0,\pi].$](http://latex.artofproblemsolving.com/b/4/c/b4ccb3ab0522405552f4590e118eccf217cce9c4.png)
We make a table of values, as shown below:
![\[\begin{array}{cccc} & x=0 & x=\frac{\pi}{2} & x=\pi \\ [1.5ex] \hline\hline \cos x & 1 & 0 & -1 \\ [1.5ex] \frac{\pi}{2}\cos x & \frac{\pi}{2} & 0 & -\frac{\pi}{2} \\ [1.5ex] f(x) & 1 & 0 & -1 \\ [1.5ex] \hline \sin x & 0 & 1 & 0 \\ [1.5ex] \frac{\pi}{2}\sin x & 0 & \frac{\pi}{2} & 0 \\ [1.5ex] g(x) & 1 & 0 & 1 \end{array}\]](http://latex.artofproblemsolving.com/d/f/3/df37a66de9d762cb9431a16adf36b11091de9cb6.png)
~MRENTHUSIASM (credit given to TheAMCHub)
Video Solution by OmegaLearn (Using Triangle Inequality & Trigonometry)
~ pi_is_3.14
See also
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 18 |
Followed by Problem 20 |
| 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 | |
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