Changes in the Cosine Curve

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. 

Please e-mail me any response you might have about this subject. 

THANKS :) 
Sabrina.

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. 

Finally, here are a few comments about C. I like to think about it 
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