I am curious if the graph of, say, y=(sinx)^2 is
considered a sinusoidal curve. That is, is a sinusoidal curve ONLY a sin
or cos curve, or is it a more generic term?
It depends upon how precise you want to be, but I think most people would
not be "offended" if you described a curve such as y = (sin x)^2 as
"sinusoidal" meaning that it had a repeating period and amplitude. I do not
think you would want to call it a "sine" curve, or "sin" curve. Those terms
most people I think would assume the standard meaning.
A sinusoidal curve is any curve that is shaped exactly like a sine or cosine
curve. It just happens to be true that y=(sinx)^2 IS shaped exactly like a
sine or cosine curve. The reason is based on a "double angle" formula:
cos(2x)=1-2*(sinx)^2. y=(sinx)^2 is the same as y=(1/2)-(1/2)cos(2x). The
curve has a different amplitude and length, but it is still shaped like a
sine function. In fact, a sine or cosine function to any power happens to
be sinusoidal. The math is more complex, but the principle is the same.
Try graphing one to see what it actually looks like.
Dr. Ken Mellendorf
Illinois Central College
On further reflection it would be more precise to refer to functions such
as: f(x) = (sin(x))^2 as "periodic functions" even though I think that in
"math-speak" sinusoidal is used loosely. What makes me rethink this is that
g(x) = tan (x) is periodic but not sinusoidal, having a periodic
discontinuity at pi/2.
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Update: June 2012