If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Sine of time

## Video transcript

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 it 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 1 over seconds or 1 over time so Omega is 1 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 1 over time is a frequency this is a constant number this is some number this is a number time is the 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 sinusoidal time increases it keeps going and going and going so now we've turned our 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 or five and that dot there that's at PI seconds and this is at 2 pi seconds right that dot right there and you can see that that is the full cycle of 1/cosine before it starts repeating again so that's six point two eight seconds so for this image here Omega has the value of one so when 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 wave 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 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 so now I want to talk a little bit more about the details of this this kind of a sinusoidal earn some new words for this so one important concept is the idea of any 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 this function and I'm going to call that that distance right there this is the period of this sine function this 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 this cosine wave this this sinusoidal II 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 2 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 we take the reciprocal of point zero two we get the answer to be that's 50 cycles per second that's the speed that's the that's the repetition rate of this sinusoidal is 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 that 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 in 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 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 going to multiply that by a conversion factor that I'm going to make up so we're going to 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 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 2pi 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 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 2pi f and f is 50 times T and that's the same as cosine of 100 by 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