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Sine of time

If we introduce time as the argument to the sine function, we get a sine wave. Animated. Created by Willy McAllister.

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  • blobby green style avatar for user Jessie Filer
    Just to confirm that I understand, are the frequency and the angular frequency the same parameter (just in different units)? For example, we could measure distance in miles or kilometers. Angular frequency just means that the frequency has different units?
    (12 votes)
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  • blobby green style avatar for user bemipefe
    We know that the frequency is the reciprocal of the period (f = 1/T and T = 1/f). In your calculation you obtain the frequency from the reciprocal of the period (1/0.02). So it seems like that 50 derives naturally from the period. Why can't it be used directly as omega value ?

    You said that frequency and angular frequency can be used interchangeably. However it seems like that the frequency in Hertz need to be always converted to rad/s in order to use it with sinusoids. So from a calculator point of view you end up using rad/s always. Assuming that a Hertz value is inserted, then the calculator need to convert it first to rad/s before performing the actual calculation. Did I get it right ?
    (2 votes)
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  • aqualine ultimate style avatar for user tantan
    what happens to the sine wave of the increasing frequency?
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      The sine function has a period of 2π. That means the sin function completes one cycle when its entire argument goes from 0 to 2π.

      ω represents the frequency of a sine wave when we write it this way: sin(ωt).

      If ω=1 the sin completes one cycle in 2π seconds.
      If ω=2π the sin completes one cycle sooner, every 1 second.

      For f>1: If ω=2πf the sin completes one cycle even faster, every 1/f seconds. Or, equivalently, sin completes f cycles in 1 second. So the higher f is, the more cycles happen in 1 second. More and more cycles get crammed into that 1 second interval. That's what higher frequency is.

      ω is called the radian frequency with units of radians per second. 2π radians is all the way around a circle (360°). 1 radian is about 57° of the way around a circle.

      f is the frequency with units of cycles per second. Cycles per second has the honorary name Hertz. ω = 2πf (good to memorize)
      (2 votes)
  • duskpin ultimate style avatar for user Thai Kingston
    What is the difference between sin and cos
    (1 vote)
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  • blobby green style avatar for user festavarian2
    At about you state that "omega is 1/time", and later you state that "omega is radians/sec". You really made this simple topic very confusing. Could you explain?
    (1 vote)
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    • old spice man green style avatar for user Willy McAllister
      The argument (the number inside) sin or cos is in radians, (a measure of the angle distance around a circle). Going around full circle is 2pi radians (of angle).

      Radians are a unit of measure of circular angles. You might think that radians have some kind of dimension. However, oddly, radians are dimensionless. Recall the definition of radians: the ratio of the distance around the circumference of a circle divided by the circle's radius. That's the ratio of two distances, like meters/meters. The resulting dimension of a radian is: none.

      If the argument of sin or cos is [\omega t] it has to be the case that the overall argument has dimension: none. That means \omega has dimension 1/t (1/time, 1/seconds, 1/minutes, whatever).

      Since we know the argument to sin or cos is a radian value, it might be more precise to include "radian" in the dimension of \omega, like this: radian/sec. This reveals why \omega is called the radian frequency.

      I perhaps made a small verbal slip by saying \omega is 1/t. It's quite common to forget the dimensionless "radians" when you are showing how \omega cancels out the seconds in t.
      (1 vote)

Video transcript

- [Voiceover] Now I wanna 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, this has to be dimensionless, this has to have no units. And 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. And 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 zero and two 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 two pi seconds, right at 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. So for this image here, omega has the value of one. So when time t reaches two pi seconds, we've gone through one full cycle. So this idea of this continuously changing cosine or sine wave going on forever, that gives us the term sine waves, and 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 rang 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 wanna do things with them. So now I wanna talk a little bit more about the details of this kind of a sinusoid wave. We're gonna 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 gonna call that, that distance right there, this is the period of this sine function. This is actually a cosine wave. The period of this 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 two T, on and on like that. And 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 can take any two points. We could actually go right here and then go through one cycle, and go to right here and I can 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 that 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. So we can say that one cycle happens every T seconds, and in our particular case, it's one cycle per 0.02 seconds, and if take the reciprocal of 0.02, we get the answer to be, that's 50 cycles per second. That's the speed, that's the repetition rate of this sinusoid, 50 cycles per second. And this has another name, it's named in honor of a German scientist, and this is called a Hertz. Heinrich Hertz is the first person to send a radio wave and receive it on purpose, he knew what he was doing. We named 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, which is measured in Hertz, and that's cycles per second. And one cycle equals two 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, and 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. It's actually sitting right here, okay. So if I give you an f, given f, what is omega? So I write down a number f, and it's in cycles per second, and I'm gonna multiply that by a conversion factor that I'm gonna make up, so we're gonna multiply that by two pi radians per second is the same as one cycle per second. And that equals, cycles per second cancels with cycles per second, so that equals two pi f radians per second. So the conversion factor is omega equals two 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 cosine two pi f times 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 'em 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 two 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.