Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Mathematics See sine curve.
 n. Anatomy Any of the venous cavities through which blood passes in various glands and organs, such as the adrenal gland and the liver.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A sine curve.
 n. Any of several channels through which venous blood passes in various organs.
 adj. sinusoidal.
 adj. That resembles a sinus.
from the GNU version of the Collaborative International Dictionary of English
 n. The curve whose ordinates are proportional to the sines of the abscissas, the equation of the curve being y = a sin x. It is also called the curve of sines.
from The Century Dictionary and Cyclopedia
 n. The curve of sines, in which the abscissas are proportional to an angle, and the ordinates to its sine.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. tiny endotheliumlined passages for blood in the tissue of an organ
 n. the curve of y=sin x
Etymologies
Examples

The reason that the sinusoid is optional is that sometimes without the sinusoid, the estimated lines end up with a wavy component.

The complex representation of a sinusoid is a*exp(j*2*pi*f*t) = a*exp(j*omega*t), and the frequency itself is f (in Hz) or omega (rad/s), which is real by definition (f=c/lambda, both of which are real numbers).

The "sinusoid" form in the autocorrelation is quite common in CRU data many other gridcells are stronger and suggests to me that the normalization has not been performed very accurately.

By randomizing the sinusoid at the bottom, I was able to achieve the effect of a curtain hanging in long folds.

To create the randomized sinusoid, I took the sum of sinusoids of random frequencies.

This wall changes smoothly from one random sinusoid at the bottom to a different one at the top.

I used a combination of horizontal and vertical sinusoids to govern the displacements, and modified my previous code to make this Manipulate, with which I could experiment with the sinusoid frequencies.

I wrote this function that uses a sinusoid to produce a gently undulating, rounded star with n points:

Huge rod is what all girls dream about! musk banshee sinusoid

Adding a sinusoid will help remove that wavy component.
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