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# Lead Lag

## Video transcript

this video we're going to 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 radiant angle of the cosine or sine now I can label these but PI over 2 represents a 90 degrees change pi is a 180 degrees this is 270 degrees and this is 360 degrees those are two equivalent scales for the the angle the angle axis in degrees or in radians now one thing we notice here is that sine a 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 that the the 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 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 could say the lead is 270 minus 180 in this case 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 and 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 waveforms changes all the time so we use the word leading 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 I do that so what I notice if I look at the value of sine right here and this is sine at 90 degrees or sine at PI over 2 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 1 but 90 degrees earlier 90 degrees before because it's a leading function so this suggests a conversion factor any time I 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 - 90 degrees if I go out to some value let's say they're on the sine curve and if I back up 90 degrees like that I'll read the same value on the cosine curve so those will give these two functions will give me the same number so I can write this identity in reverse also if I have a cosign if I'm riding along on this cosine wave what I notice is if I if I let's say I'm right here I'll notice I have the 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 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 something expressed as a sign 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've sketched on in dashed lines the negative of the orange curve so this is a negative sine wave you can see if the opposite of the the original sine wave we had so now I have the case here where the cosine is trailing or lagging the negative sign right it comes later in time right there so cosine lags negative sign and what I'll do is I'll write the same sort of identities here but in terms of this negative sign and those come out like this cosine of theta equals minus sign 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 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 some identities that are kind of useful to have around to be able to convert these these two waveforms from one to the other and back