April 7th, 2019, 03:15 AM  #1 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75  Prove inequality
Prove inequality : $\displaystyle \underbrace{\sin \sin ... \sin }_{Ntimes} (N)\leq \sin \frac{1}{N}\; $ , $\displaystyle N\in \mathbb{N}$. To write it better in shortterms : $\displaystyle \sin_{N} (N)\leq\sin \frac{1}{N}$. 
April 7th, 2019, 01:36 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,408 Thanks: 1310 
if this is a product chain it should be written as $\sin^N(N) \leq \sin\left(\dfrac 1 N \right),~N \in \mathbb{N}$ if it's a function composition chain I've seen it written as $\underset{\text{N times}}{\underbrace{\sin\circ \sin \circ \dots \sin(N)} }=\sin^{\circ N}(N) \leq \sin\left(\dfrac 1 N \right)$ which is it? 
April 7th, 2019, 01:38 PM  #3 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75 
The composition Ntimes of sin function.

April 7th, 2019, 02:00 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,265 Thanks: 690 
Wolfram Alpha gives $\sin(\sin(2)) \approx 0.78907 \dots > \frac{1}{2}$. Of course, that's in radians. However, it's $< \frac{1}{2}$ for $2$ degrees. Just wanted to note that. OP must mean degrees.
Last edited by skipjack; April 7th, 2019 at 03:21 PM. 
April 7th, 2019, 02:01 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689 
For $x \gt 0$ $\sin(x)\lt x$. Therefore $\sin(\sin(1/N))\lt \sin(1/N)$..Repeat N times to get what you want.
Last edited by skipjack; April 7th, 2019 at 03:14 PM. 
April 7th, 2019, 02:18 PM  #6 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75 
Tried all methods posted above but got no result .

April 8th, 2019, 01:18 PM  #7 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689  
April 8th, 2019, 01:19 PM  #8 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75  
April 9th, 2019, 12:11 PM  #9 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689  
April 9th, 2019, 06:06 PM  #10 
Senior Member Joined: Sep 2016 From: USA Posts: 600 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics 
Use induction and the fact that $\sin$ is entire and its Taylor expansion can be written as \[ \sin(N) = \sin(N1) + \frac{1}{2} \cos(N1)  \frac{1}{6}\sin(N1) + \dotsc \]. 

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