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
f
 (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 used by Scotsman Stuart Kenny in 1789 while observing the growth and harvest of soybeans.

The general form of the cosine function can also be d, since the cosine function is identical to the sine function (except for a horizontal shift to the left of 90º or d radians). 
Any cosine function can be written as a sine function.
d
Sinusoidal Function:
d
|A| = amplitude
B = cycles from 0 to 2π (frequency)
period = d
D = vertical shift (or displacement)
C = horizontal shift (may also be called "phase shift")

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

The number, A, in front of sine or cosine changes the height of the graph.

The value A (in front of sin or cos) affects the amplitude (height).  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:
1
2
3
aa
ss
bb
sincos16
sinso20
Notice: The graphs change "height" but do not change horizontal width.
The graphs are still drawn from 0 to 2π.

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Again, using y = A sin(Bx)   and   y = A cos(Bx).

The number, B, in front of x is number of cycles seen in 0 to 2π interval.

The value B is the number of cycles the graph completes in an interval of from 0 to 2π or 360º.  The value B affects the period.  The period of sine and cosine is g.  When 0 < B < 1, the period of the function will be greater than 2π and the graph will be a horizontal stretching.  When B > 1, the period of the function will be less than 2π and the graph will be a horizontal shrinking. 

Consider these examples:
f
d
g
a
f
dd
sinu26
sinu27
Notice: These graphs change horizontal "width" but do not change height.  The two red graphs only show us a portion (in this example "half") of the original graphs in their 0 to 2π windows.  We would need to "stretch" the domain window to 4π to see entire cycles of those two graphs.   Whereas the two blue graphs show us two complete cycles of the graphs in their 0 to 2π windows, which would allow us to "shrink" the domain window and still see complete cycles of the graphs.

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Examples:

1.  f
This problem is a combination of dealing with the values of A and B.  The A value of ½ tells us that the graph will have a vertical shrink and an amplitude of ½.  The B value of 3 tells us that 3 complete cycles of the graph will be seen in the standard domain of  0 to 2π (there will be a horizontal shrink).
The period of this new graph will be d (or 120º).

gf

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2.  a
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 ½ tells us that a complete cycle of the graph will require more than the standard domain of  0 to 2π (there will be a horizontal stretch).
The period of this new graph will be f (or 720º).

d

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3.  d
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 x-axis of the graph  y = 2 sin x.  The amplitude of 2 tells us that the graph will have a vertical stretch.

f

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ti84c
How to use your graphing calcualtor for graphing sinusoidal functions.
click here.
MathBits Calculator Pages

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