Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. 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.
 n. In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.
from the GNU version of the Collaborative International Dictionary of English
 n. The length of a perpendicular drawn from one extremity of an arc of a circle to the diameter drawn through the other extremity.
 n. The perpendicular itself. See Sine of angle, below.
 prep. Without.
from The Century Dictionary and Cyclopedia
 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 (see langsyne).
 After; since: same as since.
 To strain.
 To leave off milking a cow.
 A Latin preposition, signifying ‘without.’ See sine die, sine qua non.
 n. A gulf.
 n. 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.
 n. the function expressed by the series These functions were invented by Wronski.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. ratio of the length of the side opposite the given angle to the length of the hypotenuse of a rightangled triangle
Etymologies
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

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."

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).

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.

Whatever the merits of legislation to "reform" our health care, serious, scholarly and pragmatic evaluation, in an objective and unhurried tempo, is what we lawyers would term a sine quo non  in other words, essential.
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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
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