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### Changes in the Cosine Curve

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Date: Sat, 27 Jan 1996 17:19:07 -0500
From: Anonymous
Subject: cosine curves

Question:

I need information on the different changes in the cosine curve,
especially on the change in amplitude and the period of revolution
and phase shift of the cosine curve. I have looked through
numerous guides and reference materials on the Net, but no luck so
far.

THANKS :)
Sabrina.
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Date: Tue, 2 Jul 1996 11:41:25 -0400 (EDT)
From: Dr. Jerry
Subject: Re: cosine curves

I'll give a short answer. The reason for a short answer rests on my
opinion that the best way to understand, say, amplitude, is to start
with a brief definition and then try it out on several curves, using a
graphing calculator to test your understanding.

A general cosine curve can be written in the form y=A*cos(Bx+C),
where A, B, and C are constants. I'll make comments related to each
constant.

Since the cosine function varies between -1 and 1, the product of A
and cos(Bx+C), no matter what B and C are (B should not be zero,
to avoid trivial case), will vary between -A and A. So cosine will
vary over an interval of length |2A|. Usually, |A| is called the
amplitude.

So y=0.5cos(3x+4) has amplitude 0.5. And y=10cos(3x+4) has
amplitude 10.

The number Bx+C will increase by 2pi as x increases by an amount
(2pi)/B. For example, let's look at y when x=p and again when
x=p+(2pi)/B. We have y=A*cos(Bp+C) and
y=A*cos(B(p+(2pi)/B)+C)=A*cos(Bp+C+2pi)=A*cos(Bp+C).
So the curve starts repeating every 2pi/B radians. The period is,
then, 2pi/B.

So y=7cos(13x+27) has period 2pi/13, which is about 0.483.
And y=-3cos(x/2-0.3) has period 2pi/(1/2) = 4pi.

where the curve "starts." For example, the curve y=39cos(4x+0)
starts at x=0. This means that at x=0 we are at the top of the usual
hump of cosine.

The curve y=39cos(4x-3) starts when 4x-3=0, that is, x=3/4. At
x=3/4, the curve looks like 39cos(4x) does at the origin. So, the
curve is shifted over 3/4 units. The term phase is, I think, not
warranted. It probably comes from electrical engineering or
physics, where the cosine curve is used to model electrical circuits
and the term phase is used.

I hope this helps you.

-Doctor Jerry, The Math Forum

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Associated Topics:
High School Trigonometry

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