A
sine wave, or
sinusoid, is the graph of the sine function in trigonometry.
A sinusoid is the name given to any
curve that can be written in the form
(A and B are positive).
Sinusoids are
considered to be the general form of the sine function.
In addition to mathematics, sinusoidal functions occur in other fields of study such as science and engineering. This function also occurs in nature as seen in ocean waves, sound waves and light waves. Even average daily temperatures for each day of the year resemble this function. The term sinusoid was first use by Scotsman Stuart Kenny in 1789 while observing the growth and harvest of soybeans.
The general form of the cosine function
can also be
.
since the cosine function is identical to the sine function (except for
a horizontal shifted to the left of 90º
or radians).


Let's start with an investigation of the
simpler graphs
of
y = A sin(Bx)
and y = A cos(Bx).


The value A affects the amplitude. The amplitude (half the distance between the maximum and minimum values of the function) will be A, since distance is always positive. Increasing or decreasing the value of A will vertically stretch or shrink the graph. Consider these examples:






The value B is the number of cycles it completes in an interval of from 0 to or 360º. The value B affects the period. The period of sine and cosine is . When 0 < B < 1, the period of the function will be greater than and the graph will be a horizontal stretching. When B > 1, the period of the function will be less than and the graph will be a horizontal shrinking. Consider these examples:




Examples: 
1.
This problem is a combination of dealing with the values of A
and B. The A value of 3 tells us that the
graph will have a vertical stretch and the amplitude will be 3. The B value of 1/2 tells
us that a complete cycle of the graph will require more than the standard domain
of 0 to
(there will be a
horizontal stretch).
The period of this new graph will be
(or 720º).
2.
This problem is also a combination of dealing with the values of
A and B. The A value of 1/2 tells
us that the graph will have a vertical shrink and an amplitude of 1/2. The B value of
3 tells us that 3 complete cycles of the graph will be seen in the standard
domain of 0 to
(there will be a
horizontal shrink).
The period of this new graph will be
(or 120º).
3.
Look out for this problem. The amplitude is 2 (a positive value
representing distance). The problem may be more clearly thought of as y =
2(sin x). This graph is a reflection in the xaxis of the
graph y = 2 sin x. The amplitude of 2 tells us that
the graph will have a vertical stretch.
