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

(Added a solution)
Line 5: Line 5:
  
 
==Solution==
 
==Solution==
{{solution}}
+
<math>\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)</math>
 +
 
 +
The interval is <math>[0,\pi]</math>, so both <math>\arccos</math> and <math>\arcsin</math> will lie within the range.
 +
 
 +
<math>\frac{\pi}2 \cos x=\arcsin \left( \cos \left( \frac{\pi}2 \sin x\right)\right)</math>
 +
 
 +
<math>\frac{\pi}2 \cos x=\frac{\pi}2 - \frac{\pi}2 \sin x</math>
 +
 
 +
<math>\cos x = 1 - \sin x</math>
 +
 
 +
This only happens at <math>x = 0, \frac{\pi}2</math> on the interval <math>[0,\pi]</math>, because either <math>\cos x = 0</math> and <math>\sin x = 1</math> or <math>\cos x = 1</math> and <math>\sin x = 0</math>.
 +
 
 +
~Tucker
 +
 
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2021|ab=A|num-b=18|num-a=20}}
 
{{AMC12 box|year=2021|ab=A|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:57, 11 February 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

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

The interval is $[0,\pi]$, so both $\arccos$ and $\arcsin$ will lie within the range.

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

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

$\cos x = 1 - \sin x$

This only happens at $x = 0, \frac{\pi}2$ on the interval $[0,\pi]$, because either $\cos x = 0$ and $\sin x = 1$ or $\cos x = 1$ and $\sin x = 0$.

~Tucker


See also

2021 AMC 12A (ProblemsAnswer KeyResources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png