Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun The ordinate of the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.
 noun In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.
from The Century Dictionary.
 noun A gulf.
 noun In trigonometry, formerly, with reference to any arc of a circle, the line drawn from one extremity of the arc at right angles to the diameter which passes through its other extremity; now ordinarily, with reference not to the arc but to the angle which it subtends at the center of the circle, the ratio of the aforesaid line to the radius of the circle.
 noun the function expressed by the series These functions were invented by Wronski.
 A Latin preposition, signifying ‘without.’ See sine die, sine qua non.
 After that; afterward: same as
since , 1.  Before now; ago: same as
since , 3: as, lang syne, long ago, used also as a noun, especially in the phrase auld langsyne, old times (seelangsyne ).  After; since: same as
since .  To strain.
 To leave off milking a cow.
from the GNU version of the Collaborative International Dictionary of English.
 noun The length of a perpendicular drawn from one extremity of an arc of a circle to the diameter drawn through the other extremity.
 noun The perpendicular itself. See Sine of angle, below.
 noun logarithms of the natural sines, or logarithmic sines.
 noun See
Sinusoid .  noun the decimals expressing the values of the sines, the radius being unity.
 noun in a circle whose radius is unity, the sine of the arc that measures the angle; in a rightangled triangle, the side opposite the given angle divided by the hypotenuse. See Trigonometrical function, under
Function .  noun that part of the diameter between the sine and the arc.
 preposition Without.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun trigonometry, mathematics In a
right triangle , theratio of thelength of theside opposite anangle to the length of thehypotenuse .
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun ratio of the length of the side opposite the given angle to the length of the hypotenuse of a rightangled triangle
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

Defense counsel Paul Reichler summarized at length why the defense filed a motion to compel the government to comply with what he described as sine qua non  an essential condition  to the pretrial agreement.
Human Rights First: Al Qosi Military Commission Hearing Brings More of the Same 2010

And that's a beautifully symmetrical, simple wave that we call a sine wave.

They may also leave open the possibility of leaving the session to the 18th open all the way to that point before they call sine die, which would be final adjournment, just to make sure that no issues crop up.

"Well," said Dade, "I have heard them talking round here of a place they call a sinecuree — big pay and no work — and if there is one of them left and lying about loose I think I could fill it to a T."

"Well," said Dade, "I have heard them talking round here of a place they call a sinecuree  big pay and no work  and if there is one of them left and lying about loose I think I could fill it to a T."
Marse Henry (Volume 1) An Autobiography Henry Watterson 1880

"Well," said Dade, "I have heard them talking round here of a place they call a sinecuree  big pay and no work  and if there is one of them left and lying about loose I think I could fill it to a T."
Marse Henry, Complete An Autobiography Henry Watterson 1880

It has a frequency response from 3585 Hz (±3 dB), and the ability to deliver a shortterm sine wave of 96 dB SPL.

And these sine waves to describe the market can be long term sine waves or very short term (i.e., intraday).
Smart Trader 2008

I always wondered why a sine was called a sine, and I never would have guessed such a story.

The measurement across the plane vertically, along the line B, which is called the sine of the angle, represents the surface impact of air against the plane.
Aeroplanes 1915
Telofy commented on the word sine
Could someone enlighten me as to whether or not sine in the Latin sense of "without" is also pronounced /saɪn/? Thanks.
Edit: I hope the program is right: http://www.dict.cc/?s=sine qua non
I now have access to the online OED, I'm so ecstatic!
OED says: /'saɪnɪ/ or /'sɪneɪ/
October 6, 2008
porq commented on the word sine
The word sine defined here is the mathematical one, derived from the Latin "sinus." Most of the examplea given are uses of the unrelated Latin word "sine," meaning "without." "Sine qua non" "sine die," etc. appear a lot more frequently in literature than the mathematical term "sine." Nevettheless, there are those of us who know the difference, who have studied both mathematics and Latin, and so i would hope that this site would remove the passages that contain the Latin "sine" from this entry.
January 26, 2010