Sine of time

Now I want to introduce a new idea, and that is the idea of voltage or current, some electrical signal being a function of time: cosine of Omega T.

So here what we're doing is we're introducing time as the argument to a cosine, and time is that stuff that always goes up. This is a number that increases forever, and we have another variable in here called Omega. This is the Greek lowercase Omega, and Omega has an important job in this. The argument to cosine, whatever is inside the cosine, has to be dimensionless; this has to have no units.

So if we put in a unit of seconds, that means Omega is something that has the units of one over seconds, or one over time. So Omega is one over time, and when we multiply those two numbers together, we get something that has no units, and then we can take the cosine of it.

So this is referred to as a frequency. Something that has units of one over time is a frequency; this is a constant number. This is some number; this is a number time is a number that increases all the time. And so when we have that cosine, we now have something we call a cosine wave or a sine wave or a sinusoid.

That sine wave goes on. As time increases, it keeps going and going and going. So now we've turned our trigonometric cosine function, which is right here, which is something that was well defined between 0 and 2 pi radians. Notice that I've changed the axis; the axis is now in time over here, and now we're counting off time in seconds.

There's two seconds, three, four, five, and that dot there, that's at Pi seconds, and this is at 2 pi seconds. Write that dot right there, and you can see that that is the full cycle of one cosine before it starts repeating again. So that's 6.28 seconds.

For this image here, Omega has the value of one. So when time T reaches 2 pi seconds, we've gone through one full cycle. So this idea of this continuously changing cosine or sine wave going on forever gives us the term sine waves. Sine waves are a good model for a lot of things that happen in nature.

If you ever hear a pure tone or a pure note, a bell being rung, or a whistle, or if you sing a note, the shape of those tones looks like a sine wave or a cosine wave, and these are often the things that come into our electronic systems, and we want to do things with them.

Now I want to talk a little bit more about the details of this kind of a sinusoid wave. We're going to make up; we're going to learn some new words for this. So one important concept is the idea of any repeating waveform; any repeating signal is the idea of a period.

Let's just do the zero crossings here. If I take the time change from there to there, this is the repeating interval of this function, and I'm going to call that distance right there. This is the period of this function; this is actually a cosine wave. The period of the sine is this distance in time right here, and the symbol we use is typically a capital T to indicate the period.

So let's look at this cosine wave, this sinusoid, and identify what its period is. I can do it easily if I go right here. It looks like it repeats on this interval right here every time we hit one of those points.

So this would be, if this is time zero here, this is time Big T, this is time 2T, and on and on like that. I read off this graph, and what I see right here is that the time is T equals 0.02 seconds. So that's how you find out what the period of something is.

You could take any two points; we could actually go right here and then go through one cycle and go to right here, and I could read off that period there. There's T, and that's the same value as that T right over there. So the time T, we can also call it a cycle; that's the time it takes to go through one period.

That's one cycle. So one of the questions I can ask about this waveform is how many cycles fit in one second. How many cycles per second is another way to say that? We can say that one cycle happens every T seconds, and in our particular case, it's one cycle per 0.02 seconds.

If we take the reciprocal of 0.02, we get the answer to be 50 cycles per second. That's the speed; that's the repetition rate of this, 50 cycles per second. This has another name. It's named in honor of a German scientist, and this is called a Hertz.

Hinrich Hertz is the first person to send a radio wave and receive it on purpose. He knew what he was doing. We name the unit cycles per second in his honor, and that's called the Hertz.

So now we have two ways to measure frequency. One is F, which is frequency, measured in hertz, and that's cycles per second. One cycle equals 2 pi radians per second. So the two measures are cycles per second and radians per second, and we'll flip back and forth between those.

Okay, and radians per second or the variable is Omega, and that's called angular frequency or radian frequency. You'll sometimes see the word rad used to indicate that we're talking about angular or radian frequency, and the variable is Omega.

So let's work out what the relationship between F and Omega is. It's actually sitting right here. Okay, so if I give you an F, give an F, what is Omega? So I write down a number F, and it's in cycles per second, and I'm going to multiply that by a conversion factor that I'm going to make up.

So we're going to multiply that by 2 pi radians per second, which is the same as one cycle per second, and that equals cycles per second cancels with cycles per second, so that equals 2 pi F radians per second.

So the conversion factor is Omega equals 2 pi F, and that's worth remembering. So if I have a sine wave, a voltage sine wave, for instance, V of T equals cosine Omega T, I can write that equivalently as V of T equals sine 2 pi f * T.

So one of the frequencies, this one, is in cycles per second, and this one is in radians per second, and we can interchange them that way using this conversion factor. So if we take the example from earlier in the video, we had a signal that was 50 hertz or 50 cycles per second.

So we would write that here like this: we'd say V of T equals cosine 2 pi f, and F is 50 times T, and that's the same as cosine of 100 Pi T. So this number right here, 100 Pi, that's Omega, and this number right here is F.

So that does it for our review of trigonometry, and we've introduced the idea of a sine wave where T is the argument to the trig function.