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# Sine and cosine come from circles

## Video transcript

now I'm going to clear off the screen here and we're going to talk about the shape of the sine function let's do that this is a plot of the sine function where the angle theta this is the theta axis in this plot where where theta has been plotted out on a straight line instead of wrapped around this circle so if we draw a line on here let's make this this circle of radius one so if I draw this line up here and it's on a unit circle the definition of sine of theta this will be theta here is opposite over hypotenuse so this is the the opposite side and that distance is the opposite leg of that triangle is this value right here so sine of theta is actually equal to Y over the hypotenuse and the hypotenuse is one in all cases around this so if I plot this on a curve this is an angle and I basically go over here and plot it like that and then as as theta swings around the circle I'm going to plot the different values of Y if it comes over this way down here like this right you can see that that plots over there like that now when the angle gets back all the way to zero of course the sine function comes all the way back to zero and then it repeats again as our vector swings around the other way so the sine of two pi is zero just like the sine of zero so every two pi if I go off the screen every 2 pi comes back and repeats to zero so now I want to do the same thing with the cosine function that we did was sign where we project the projection of this value onto this time the cosine curve down here this has the cosine curve with time going down on the page and our definition of cosine was adjacent over hypotenuse hypotenuse is one in our drawing so cosine of theta equals adjacent which is X the x value divided by hypotenuse which is one so in this diagram the cosine of theta is actually the x value which is this X right here so let me clean this off for a second and we'll start at the beginning let's start with the radius pointing straight sideways and we know that cosine of theta equals zero is 1 so if I drop that down if I project that down on to the angle 0 that's this point right here on the curve now as we roll forward we go to a higher angle this projection now moves to here on the curve when the arrow is straight up we are at this point right here we go back through the axis if we go continue on this projects down here and we're moving this radius vector around in a circle like this eventually this one will be at the same point as before as the one above but it'll be on this part of the curve here and when we get back to zero again the projection is to this point here so that's a way to visualize the cosine curve getting generated by a vector rotating around the circle the cosine comes out the bottom because it's the projection on the x-axis and when we did the sine it was the projection on the y-axis produce the sine wave when we went this way so I like to visualize this because this this rotating vector is a really simple and powerful idea and we can see how it actually generates it's a way to generate sines and cosine waves and you can see how sort of naturally they come out at different phases right the sine starts at zero and the cosine starts at one with this way of drawing it you can see why that happens so this relationship between circles and rotating vectors and sines and cosines is a very powerful idea we're really going to take advantage of this