Changes in the Cosine CurveDate: 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 |
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