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Period and frequency of trig functions
By Fireblase on March 24, 1998:

Could you please answer my question with all the working out, so I can understand this.

Write down in terms of p, the period and frequency of the following waves:
4Cos(2t) and 5sin(4t + p/6)

By David Sanders on March 24, 1998:

Dear Fireblase,

To answer this question, we need to think about what frequency and period really mean. I'll consider period first.

The period of a wave is the time it takes for the wave to repeat. If you draw a graph of it, it's the horizontal distance you have to shift the wave along before it looks the same.

At this stage, I recommend that you draw graphs of the waves you've talked about. You don't need to make them too accurate, just sketch what they look like. If you've got a graphical calculator or graph drawing program on a computer, you could do it on that, but only AFTER you've tried to do it yourself!

Firstly, look at those factors in front of the sin or cos. Draw graphs of y = 4 cos(t) and y = cos (t).

You should notice that one is 4 times as tall as the other, but looks just the same apart from that - it's just been stretched 4 times in the vertical direction. So the 4 (or the 5) don't affect the period. What they do affect is the AMPLITUDE (the size) of the wave. The amplitude of y = 4 cos(t) is 4, whereas the amplitude of y = cos(t) is 1. You can see waves of different amplitude by looking at a river - some waves look very small - they have small amplitude - and others are bigger. It turns out that waves with bigger amplitude carry more energy, but that's just by the way. If you are studying or are going to study A-level physics, you'll probably learn about that.

So now we've learnt that we can ignore those factors in front of the sin or cos, for our present purposes at least. What about the factors INSIDE the sin or cos? What is the difference between y = cos(t) and y = cos(2t)?

Once again, you should draw graphs of both - on the same axes - and compare.

You will see that y = cos(2t) oscillates faster than y = cos(t). If you go across by the same amount, cos(2t) completes more cycles than cos(t) does.

Why is this? Well, what is the period of cos(t)? The cos graph repeats itself once every 360 degrees, in other words 2p radians. cos(t + 2p) = cos(t), for any t. You can see this from the graph, or if you know the addition formula, cos(A+B) = cos(A)cos(B) - sin(A)sin(B), you can plug in A=t, B=2p, and use cos(2p)=1, sin(2p)=0. (If you don't know this, don't worry, just look at the graph again.)

Now, as you move along the cos graph, it takes 2*p to get back to where you started. Now look at the cos(2t) graph. If you move along just p, then 2t has moved through 2*p, and so again you're back where you started! Think about this carefully - this is the crux of the matter. But although 2t has changed by 2p, t has only changed by p. Thus the period of cos(2t) is p.

And by my argument above about amplitudes, the answer to the first question is that 4cos(2t) has period p, too.

I hope from this argument that you can work out what the period of cos(4t) is.

Now draw a graph of cos(t+p/6), on top of one of cos(t). I hope you can see that it's just been shifted along a bit (p/6, in fact). But this doesn't change its period, because the period is just how much you have to shift it along by to get it back to the start - this isn't changed by shifting the whole lot along! So the period of cos(t+p/6) is the same as that of cos(t).

Now draw a graph of sin(t) and one of cos(t) on the same graph. You should be able to see that sin(t) is just the same graph, but again shifted along. How much is it shifted along by?

Put this all together and you should have an answer to your second question. I know you asked for full working, but explanations are much more helpful! If you're really stuck, then ask again.

Hope this has helped,

David.

By David Sanders on Thursday, September 7, 2000 - 01:39 pm:

Sorry, I forgot to tell you about the frequency!

Well, the frequency is how many waves pass in a unit time interval - normally 1 second.

Suppose the period is 1 second. Then one wave passes each second, so the frequency is 1 wave per second. If the period is 1/2 a second, then 2 waves pass every second, so the frequency is 2. What about if the period is 2 seconds?

From this, you should be able to work it out for a wave with any period!

David.