Mixing
$| sin x| > frac{sqrt{2-sqrt{2}}}{2}$ iff $1/8< { frac{x}{pi}} < 7/8$ where ${x}$ is the fractional...
$begingroup$
This is a problem that arose while reading the book "Putnam and Beyond":
Why is $| sin x| > frac{sqrt{2-sqrt{2}}}{2}$ iff $frac18 < { frac{x}{pi}} < frac78$ where ${x}$ is the fractional part of $x?$
This was used in proving the divergence of $sum frac{ |sin n|}{n}$, although the divergence is not in question here.
trigonometry inequality problem-solving fractional-part
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
$endgroup$
add a comment |
$begingroup$
This is a problem that arose while reading the book "Putnam and Beyond":
Why is $| sin x| > frac{sqrt{2-sqrt{2}}}{2}$ iff $frac18 < { frac{x}{pi}} < frac78$ where ${x}$ is the fractional part of $x?$
This was used in proving the divergence of $sum frac{ |sin n|}{n}$, although the divergence is not in question here.
trigonometry inequality problem-solving fractional-part
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
$endgroup$
$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
1
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14
add a comment |
$begingroup$
This is a problem that arose while reading the book "Putnam and Beyond":
Why is $| sin x| > frac{sqrt{2-sqrt{2}}}{2}$ iff $frac18 < { frac{x}{pi}} < frac78$ where ${x}$ is the fractional part of $x?$
This was used in proving the divergence of $sum frac{ |sin n|}{n}$, although the divergence is not in question here.
trigonometry inequality problem-solving fractional-part
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
$endgroup$
This is a problem that arose while reading the book "Putnam and Beyond":
Why is $| sin x| > frac{sqrt{2-sqrt{2}}}{2}$ iff $frac18 < { frac{x}{pi}} < frac78$ where ${x}$ is the fractional part of $x?$
This was used in proving the divergence of $sum frac{ |sin n|}{n}$, although the divergence is not in question here.
trigonometry inequality problem-solving fractional-part
trigonometry inequality problem-solving fractional-part
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
edited Feb 2 at 13:50
Michael Lugo
18.4k33576
18.4k33576
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
asked Jan 27 at 1:45
Cute BrownieCute Brownie
1,085417
1,085417
$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
1
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14
add a comment |
$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
1
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14
$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
1
1
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14
add a comment |
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$begingroup$
Just to clarify, you mean ${x}=x-lfloor xrfloor$, not ${x}=xleft(x-lfloor xrfloorright)$, right?
$endgroup$
– Carl Schildkraut
Feb 2 at 6:31
1
$begingroup$
$frac{sqrt{2-sqrt{2}}}2=sin frac{pi}8$
$endgroup$
– random
Feb 4 at 1:14