Difference between revisions of "2021 AMC 12A Problems/Problem 19"

Revision as of 19:29, 25 March 2021

Problem

How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$ ?

$\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Solution 1 (Inverse Trigonometric Functions)

$\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$

The ranges of $\frac{\pi}2 \sin x$ and $\frac{\pi}2 \cos x$ are both $\left[-\frac{\pi}2, \frac{\pi}2 \right]$ , which is included in the range of $\arcsin$ , so we can use it with no issues.

$\frac{\pi}2 \cos x=\arcsin \left( \cos \left( \frac{\pi}2 \sin x\right)\right)$

$\frac{\pi}2 \cos x=\arcsin \left( \sin \left( \frac{\pi}2 - \frac{\pi}2 \sin x\right)\right)$

$\frac{\pi}2 \cos x=\frac{\pi}2 - \frac{\pi}2 \sin x$

$\cos x = 1 - \sin x$

$\cos x + \sin x = 1$

This only happens at $x = 0, \frac{\pi}2$ on the interval $[0,\pi]$ , because one of $\sin$ and $\cos$ must be $1$ and the other $0$ . Therefore, the answer is $\boxed{C) 2}$

~Tucker

Solution 2 (Cofunction Identity)

By the cofunction identity $\sin\theta=\cos\left(\frac{\pi}{2}-\theta\right)$ for all $\theta,$ we simplify the given equation: $\begin{align*} \sin \left( \frac{\pi}2 \cos x\right) &= \cos \left( \frac{\pi}2 \sin x\right) \\ \cos \left(\frac{\pi}2-\frac{\pi}2 \cos x\right) &= \cos \left( \frac{\pi}2 \sin x\right) \\ \cos \left(\frac{\pi}2 \left(1 - \cos x \right)\right) &= \cos \left( \frac{\pi}2 \sin x\right) \\ \frac{\pi}2 \left(1 - \cos x \right) &= \frac{\pi}2 \sin x + 2n\pi, \end{align*}$ for some integer $n.$ We keep simplifying: $\begin{align*} 1 - \cos x &= \sin x + 4n \\ 1 - 4n &= \sin x + \cos x. \end{align*}$ By rough constraints, we know that $-2 < \sin x + \cos x < 2,$ so that $-2 < 1 - 4n < 2.$ The only possibility is $n=0.$ From here, we get $\begin{align*} \sin x + \cos x &= 1 \ \ \ \ \ (*) \\ \sin^2 x + \cos^2 x + 2\sin x \cos x &= 1 \\ 2\sin x \cos x &= 0 \\ \sin(2x) &= 0 \\ 2x &= k\pi \\ x &= \frac{k\pi}{2}, \end{align*}$ for some integer $k.$

The possible solutions in $[0,\pi]$ are $x=0,\frac{\pi}{2},\pi,$ but only $x=0,\frac{\pi}{2}$ check the original equation (Note that $x=\pi$ is an extraneous solution formed by squaring $(*)$ above.). Therefore, the answer is $\boxed{\textbf{(C) }2}.$

~MRENTHUSIASM

Solution 3 (Graphs and Analysis)

Let $f(x)=\sin\left(\frac{\pi}{2}\cos x\right)$ and $g(x)=\cos \left( \frac{\pi}2 \sin x\right).$ This problem is equivalent to counting the intersections of the graphs of $f(x)$ and $g(x)$ in the closed interval $[0,\pi].$ We make a table of values, as shown below: $\begin{array}{c|ccc} & & & \\ [-2ex] & \boldsymbol{x=0} & \boldsymbol{x=\frac{\pi}{2}} & \boldsymbol{x=\pi} \\ [1.5ex] \hline & & & \\ [-1ex] \boldsymbol{\cos x} & 1 & 0 & -1 \\ [1.5ex] \boldsymbol{\frac{\pi}{2}\cos x} & \frac{\pi}{2} & 0 & -\frac{\pi}{2} \\ [1.5ex] \boldsymbol{f(x)} & 1 & 0 & -1 \\ [1.5ex] \hline & & & \\ [-1ex] \boldsymbol{\sin x} & 0 & 1 & 0 \\ [1.5ex] \boldsymbol{\frac{\pi}{2}\sin x} & 0 & \frac{\pi}{2} & 0 \\ [1.5ex] \boldsymbol{g(x)} & 1 & 0 & 1 \\ [1ex] \end{array}$ For the graphs of $f(x)$ and $g(x),$ we will analyze their increasing/decreasing behaviors in $[0,\pi]:$

The graph of $f(x)$ in $[0,\pi]$ (from left to right) has the same behavior as the graph of $y=\sin x$ in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ (from right to left): the output is from $1$ to $-1$ (from left to right), inclusive, and strictly decreasing.

The graph of $g(x)$ in $[0,\pi]$ (from left to right) has two parts:

The graph of $g(x)$ in $\left[0,\frac{\pi}{2}\right]$ has the same behavior as the graph of $y=\cos x$ in $\left[0,\frac{\pi}{2}\right]$ (from left to right): the output is from $1$ to $0$ (from left to right), inclusive, and strictly decreasing.

The graph of $g(x)$ in $\left[\frac{\pi}{2},\pi\right]$ has the same behavior as the graph of $y=\cos x$ in $\left[0,\frac{\pi}{2}\right]$ (from right to left): the output is from $0$ to $1$ (from left to right), inclusive, and strictly increasing.

If $x\in\left(\frac{\pi}{2},\pi\right],$ then $f(x)<0$ and $g(x)>0.$ So, their graphs do not intersect.

If $x\in\left[0,\frac{\pi}{2}\right],$ then $0\leq f(x),g(x)\leq1.$ Clearly, the graphs intersect at $x=0$ and $x=\frac{\pi}{2}$ (at points $(0,1)$ and $\left(\frac{\pi}{2},0\right),$ respectively), but we will prove or disprove that they are the only points of intersection:

Let $A=\frac{\pi}{2}\cos x$ and $B=\frac{\pi}{2}\sin x.$ It follows that $A,B\in\left[0,\frac{\pi}{2}\right].$ Since $\sin A = \cos B,$ we know that $A+B=\frac{\pi}{2}$ by the cofunction identity: $\begin{align*} \frac{\pi}{2}\cos x + \frac{\pi}{2}\sin x &= \frac{\pi}{2} \\ \cos x + \sin x &=1. \end{align*}$

We apply Solution 2's argument (starts from its last block of equations) to deduce that $(0,1)$ and $\left(\frac{\pi}{2},0\right)$ are the only points of intersection. So, the answer is $\boxed{\textbf{(C) }2}.$

Graphs of $f(x)$ and $g(x)$ in Desmos: https://www.desmos.com/calculator/g6jvjnubrb

~MRENTHUSIASM (credit given to TheAMCHub)

Video Solution by OmegaLearn (Using Triangle Inequality & Trigonometry)

https://youtu.be/wJxN1YPuyCg

~ pi_is_3.14

@@ Line 72: / Line 72: @@
 *The graph of <math>g(x)</math> in <math>[0,\pi]</math> (from left to right) has two parts:
-** The graph of <math>g(x)</math> in <math>\left[0,\frac{\pi}{2}\right]</math> has the same behavior as the graph of <math>y=\cos x</math> in <math>\left[0,\frac{\pi}{2}\right]</math> (from left to right): the output is from <math>1</math> to <math>0</math> (from left to right), inclusive, and strictly decreasing.
+<ol style="margin-left: 3em;">
-** The graph of <math>g(x)</math> in  <math>\left[\frac{\pi}{2},\pi\right]</math> has the same behavior as the graph of <math>y=\cos x</math> in <math>\left[0,\frac{\pi}{2}\right]</math> (from right to left): the output is from <math>0</math> to <math>1</math> (from left to right), inclusive, and strictly increasing.
+  <li>The graph of <math>g(x)</math> in <math>\left[0,\frac{\pi}{2}\right]</math> has the same behavior as the graph of <math>y=\cos x</math> in <math>\left[0,\frac{\pi}{2}\right]</math> (from left to right): the output is from <math>1</math> to <math>0</math> (from left to right), inclusive, and strictly decreasing.</li><p>
+  <li>The graph of <math>g(x)</math> in  <math>\left[\frac{\pi}{2},\pi\right]</math> has the same behavior as the graph of <math>y=\cos x</math> in <math>\left[0,\frac{\pi}{2}\right]</math> (from right to left): the output is from <math>0</math> to <math>1</math> (from left to right), inclusive, and strictly increasing.</li><p>
+</ol>
 If <math>x\in\left(\frac{\pi}{2},\pi\right],</math> then <math>f(x)<0</math> and <math>g(x)>0.</math> So, their graphs do not intersect.

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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