Lead Lag

In this video, we're going to introduce a couple of words to help talk about the relationship between sine and cosine, or different sinusoids that have the same frequency but a different timing relationship.

So what I've shown here is a plot of a cosine and a sine wave, and the axis here is in theta, in the angle, the radian angle of the cosine or sine. Now I can label these: π/2 represents a 90° change, π is a 180°, this is 270°, and this is 360°. Those are two equivalent scales for the angle axis, in degrees or in radians.

Now, one thing we notice here is that sine and cosine look the same, but they don't overlap. If I change this to a time axis, what I can say is that the cosine wave reaches its peak at time equals z, and the sine wave reaches its peak at a later time. This is increasing time going this way, so the sine is delayed compared to the cosine. The peak here is delayed here.

If I go down and look at these two peaks, we see the same relationship. This sine peak in orange is behind, is delayed from the cosine. So when we have this timing relationship between two periodic waves, what we say is, in this case, that the cosine leads the sine wave. The amount of lead is the difference between these two points, and we could say the lead is 270 minus 180. In this case, it would be 90°.

So we say that cosine leads sine by 90°. Now I can take exactly the opposite point of view. If I actually measure where the sine is relative to cosine and I say it's behind, then we would say it lags. So the phrase we hear would be sine lags cosine by 90°.

So that's the term lead and lag; that's what those mean. Now these terms apply, this idea of a delay, this only applies when these frequencies are the same. If the frequencies are different, the relationship between the two waveforms changes all the time. So we use the word lead and lag when we know that the two signals we are talking about are exactly the same frequency.

One thing I want to be able to do is express sine and cosine in terms of each other. So if I have a sine wave, could I actually express this orange curve as a cosine wave? How would I do that?

What I notice, if I look at the value of sine right here, and this is sine at 90° or sine at π/2, if I look at this value here, what I notice is that this has the same value; the peak of one cosine has the same value as the peak of sine, but 90° earlier, 90° before because it's a leading function.

So this suggests a conversion factor. Anytime I pick out a value of sine, if I look back 90°, I'll see the same value for cosine. So I can write something like this: I can say that sine of theta equals cosine of theta minus 90°.

If I go out to some value, let's say there on the sine curve, and if I back up 90°, like that, I'll read the same value on the cosine curve. So these two functions will give me the same number. I can write this identity in reverse also. If I have a cosine, if I'm writing along this cosine wave, what I notice is if I, let's say I'm right here, I'll notice I have my peak value here.

If I added, if I went later in time or if I added 90°, I would have the same value on that orange sine curve. So if I look here on cosine, I want to know what that is in terms of a sine function. If I add 90° to the argument, the sine function will give me the same value.

So what that says is cosine theta equals sine of theta plus 90°. These are two identities; we can use this to convert something expressed as a sine into a cosine or vice versa.

Now I want to show you two more identities that are actually pretty useful. Here, what I have is I've sketched on dashed lines the negative of the orange curve. So this is a negative sine wave; you can see it's the opposite of the original sine wave we had.

Now I have the case here where the cosine is trailing or lagging the negative sine. Right? It comes later in time, right there. So cosine lags negative sine. What I'll do is I'll write the same sort of identities here, but in terms of this negative sine, and those come out like this: cosine of theta equals minus sine of theta minus 90°.

So what that means is if I want to know the value of cosine, I can flip that around the same way, and I can say that negative sine of theta equals cosine of theta plus 90°. That's the same identity but in reverse. If I want to know the value of negative sine, I just take that argument, I add it, I advance it by 90°, and take the cosine; it'll have the same value.

So this identity and this identity are pretty useful to have around. This one allows us to convert sine and cosine together. This pair here is useful for moving negative signs around.

So that's what I wanted to say about lead and lag. These are sort of slang or jargon, the nicknames of the relationship between two different waveforms of the same frequency but different phase timing, different phase delay. And then we worked out some identities that are kind of useful to have around to be able to convert these two waveforms from one to the other and back.