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# Lead Lag

## Video transcript

- [Voiceover] In this video we're gonna introduce a couple of words to help talk about the relationship between sines and cosines, 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, the pi over two represents a 90 degrees change. Pi is 180, degrees. This is 270 degrees. And this is 360 degrees. Those are two equivalent, scales for the angle, the angle axis, in degrees or in radians. Now, one thing we notice here is that sine and cosine, they look the same, but they don't overlap. If I change this to a time, If I change this to a time, axis, what I can say is the cosine wave reaches its peak at time equals zero, and the sine peak, 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, we say that the cosine, leads, the sine wave. And the amount of lead is the difference between these two points, and we can say the lead is 270 minus 180, in this case, it'd be 90 degrees. So we say that cosine, leads sine by 90 degrees. Now I can take exactly the opposite point of view. If I actually measure where the sine is relative to cosine, I say it's behind, then we would say it lags. So the phrase we hear would be sine lags cosine by 90 degrees. 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 wave functions 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 sines and cosines 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 a do that? So what I've noticed, if I look at the value of sine right here, and this is sine at 90 degrees, or sine at pi over two, if I look at this value here, what I notice is that, this has the same value, which is the peak of one. Cosine has the same value, the peak of one, but 90 degrees earlier, 90 degrees before, because it's a leading function. So this suggests a conversion factor. Any time I pick out a value of the sine, if I look back 90 degrees, I'll see the same value for cosine. So I can write something like this, I can say that sine of theta equals the cosine of theta, minus 90 degrees. If I go out to some value, let's say there on the sine curve, and if I back up 90 degrees, like that, I'll read the same value on the cosine curve, so these two functions will give me the same number. So I can write this identity in reverse also. If I have a cosine, if I'm riding along on this cosine wave, what I notice is, if let's say I'm right here, I'll notice I have my peak value here, and if I added, if I went later in time, or if I added 90 degrees, I would have the same value on that orange sine curve. So if I look here, on cosine, if I want to know what that is, in terms of a sine function, if I add 90 degrees 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. So these are two identities, we can use this to convert a 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. And here what I have is, I sketched on in dash lines the negative of the orange curve, so this is a negative sine wave. You can see it the opposite of the original sine wave we had. So 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. And 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, so that's the dashed one, of theta, minus 90 degrees. So what that means, if I want to know the value of cosine, and 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 degrees, 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 sines and cosines together. This pair here is useful for moving minus sines 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 wave forms of the same frequency but different phase timing, different phase delay. And then we worked out some, some identities that are kind of useful to have around, to be able to convert these two wave forms from one to the other and back.