April 7th, 2019, 03:15 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 540 Thanks: 82  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: 2,467 Thanks: 1342 
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: somewhere Posts: 540 Thanks: 82 
The composition Ntimes of sin function.

April 7th, 2019, 02:00 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,329 Thanks: 720 
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: 6,770 Thanks: 700 
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: somewhere Posts: 540 Thanks: 82 
Tried all methods posted above but got no result .

April 8th, 2019, 01:18 PM  #7 
Global Moderator Joined: May 2007 Posts: 6,770 Thanks: 700  
April 8th, 2019, 01:19 PM  #8 
Senior Member Joined: Dec 2015 From: somewhere Posts: 540 Thanks: 82  
April 9th, 2019, 12:11 PM  #9 
Global Moderator Joined: May 2007 Posts: 6,770 Thanks: 700  
April 9th, 2019, 06:06 PM  #10 
Senior Member Joined: Sep 2016 From: USA Posts: 624 Thanks: 396 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 \[ \sin(N) = \sin(N1) + \frac{1}{2} \cos(N1)  \frac{1}{6}\sin(N1) + \dotsc \]. 

Tags 
inequality, prove 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Prove an inequality.  rubis  Elementary Math  0  October 11th, 2018 05:05 AM 
Triangle Inequality: Prove Absolute Value Inequality  StillAlive  Calculus  5  September 2nd, 2016 11:45 PM 
How do you prove this inequality?  davedave  Algebra  3  April 26th, 2014 05:19 PM 
prove an inequality  Albert.Teng  Algebra  2  April 9th, 2013 05:32 AM 
prove log inequality  Albert.Teng  Algebra  11  July 13th, 2012 05:42 AM 