2021 AMC 12A Problems/Problem 19
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Contents
[hide]Problem
How many solutions does the equation have in the closed interval
?
Solution 1 (Inverse Trigonometric Functions)
The ranges of and
are both
, 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
We make a table of values, as shown below:
~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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