MHB Can You Prove $\cos(\cos 1) > \sin(\sin(\sin 1))$?

  • Thread starter Thread starter anemone
  • Start date Start date
AI Thread Summary
The discussion centers on proving the inequality $\cos(\cos 1) > \sin(\sin(\sin 1))$. Participants are encouraged to engage with the Problem of the Week (POTW) format, which emphasizes mathematical rigor and community involvement. Previous POTWs have seen low engagement, as noted by the lack of responses to the last problem. The thread also references guidelines for participation and submission of solutions. Overall, the focus remains on solving the current mathematical challenge presented.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Prove $\cos (\cos 1) > \sin (\sin (\sin 1))$.

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
No one answered last week's POTW(Sadface), but you can find the suggested solution as follows:

For $x>0$, we have

$1-\dfrac{x^2}{2}< \cos x<1-\dfrac{x^2}{2}+\dfrac{x^4}{24}$

When $x=1$, we see that

$\cos 1<1-\dfrac{1^2}{2}+\dfrac{1^4}{24}=\dfrac{13}{24}$

Since $\cos x$ is decreasing. it follows that

$\cos (\cos 1)>\cos \left(\dfrac{13}{24}\right)>1-\dfrac{1}{2}\left(\dfrac{13}{24}\right)^2>0.85$

Next, recall that $\sin x<x$ for $x>0$.

Since $\sin x$ is increasing, we have

$\sin (\sin (\sin 1))<\sin (\sin 1)< \sin 1$

But we also know that

$\sin x<x-\dfrac{x^3}{6}+\dfrac{x^5}{120}$ for $x>0$.

It follows that

$\sin (\sin (\sin 1))< \sin 1<\dfrac{101}{120}<0.85<\cos (\cos 1)$ (Q.E.D.)
 
Back
Top