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0

$begingroup$

I would like to find tight bounds of the form

$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$

Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.

For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.

real-analysis calculus

share|cite|improve this question

edited Mar 11 at 20:18

Sebastian Schlecht

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
  $endgroup$
  – mfl
  Mar 11 at 20:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @mfl as tight as possible bounds for the given values.
  $endgroup$
  – Sebastian Schlecht
  Mar 11 at 20:06
  

  
  
  
  

add a comment | 

0

$begingroup$

I would like to find tight bounds of the form

$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$

Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.

For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.

real-analysis calculus

share|cite|improve this question

edited Mar 11 at 20:18

Sebastian Schlecht

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
  $endgroup$
  – mfl
  Mar 11 at 20:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @mfl as tight as possible bounds for the given values.
  $endgroup$
  – Sebastian Schlecht
  Mar 11 at 20:06
  

  
  
  
  

add a comment | 

$begingroup$

I would like to find tight bounds of the form

$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$

Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.

For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.

real-analysis calculus

share|cite|improve this question

edited Mar 11 at 20:18

Sebastian Schlecht

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

$endgroup$

share|cite|improve this question

edited Mar 11 at 20:18

Sebastian Schlecht

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

share|cite|improve this question

edited Mar 11 at 20:18

Sebastian Schlecht

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718

asked Mar 11 at 19:59

Sebastian SchlechtSebastian Schlecht

23718