2021 AMC 12A Problems/Problem 19
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
Solution 2 (Cofunction Identity)
By the cofunction identity for all we simplify the given equation: for some integer We keep simplifying: By rough constraints, we know that from which The only possibility is From here, we get for some integer
The possible solutions in are but only check the original equation (Note that is an extraneous solution formed by squaring above.). Therefore, the answer is
Solution 3 (Graphs and 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: For the graphs of and we will analyze their increasing/decreasing behaviors in
- The graph of in (from left to right) has the same behavior as the graph of in (from right to left): the output is from to (from left to right), inclusive, and strictly decreasing.
- The graph of in (from left to right) has two parts:
- The graph of in has the same behavior as the graph of in (from left to right): the output is from to (from left to right), inclusive, and strictly decreasing.
- The graph of in has the same behavior as the graph of in (from right to left): the output is from to (from left to right), inclusive, and strictly increasing.
If then and So, their graphs do not intersect.
If then Clearly, the graphs intersect at and (at points and respectively), but we will prove or disprove that they are the only points of intersection:
Let and It follows that Since we know that by the cofunction identity:
We apply Solution 2's argument (starts from its last block of equations) to deduce that and are the only points of intersection. So, the answer is
~MRENTHUSIASM (credit given to TheAMCHub)
The graphs of (in red) and (in blue) are shown below.
Graph in Desmos: https://www.desmos.com/calculator/brjh3gybcc
Video Solution by OmegaLearn (Using Triangle Inequality & Trigonometry)
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