Sine and cosine from rotating vector

now I'd like to demonstrate one way to construct a sine wave what we're going to do is we're going to construct something that looks like sine of Omega T so we have our function of time here and we have our frequency now this little animation is going to show us a way to construct a sine wave so what I have here this green line is a rotating vector and let's just say that the radius of this circle is one so here's a vector just rotating slowly around and around and in the dotted line here in that yellow dot going up and down that's the projection that's the projection of the tip of the green arrow onto the y-axis and as the vector goes round and around you can see that the projection on the y-axis is bobbing up and down and up and down and that's actually going up and down in a sine wave pattern so now I'm going to switch to a new animation and we'll see we'll see what that dot looks like as it goes up and down in time so here's the plot here's what a sine wave looks like as you notice when the green line goes through zero right there let's wait till it comes around again the value of the yellow line when it goes through zero is zero so this yellow line here is a plot of sine of Omega T now if I go to a projection this projection was onto the y-axis and I can do the same animation but this time project on to the x-axis and that'll produce for us a cosine wave let's see what that looks like now in this case if we switched over you can see that the projection that dotted green line is on to the x-axis and what this is doing is it's producing a cosine wave for us so this is this is going to be cosine of Omega T now because we're tracking the progress on the x-axis the cosine wave seems to emerge going down on the page so the time axis is down here when the green arrow is zero right there the value of the cosine was one and when it's minus 180 degrees it's minus one on the cosine so that's why this is a cosine wave and it has the same frequency as the sine wave we generated and I want to show you these two together because it's just sort of a beautiful drawing you I'll leave our animation here for a second we sine-wave being generated in yellow and in orange we see the cosine wave being generated and they're both coming from this rotating green vector so this is a really simple demonstration of a way to generate sines and cosines with this rotating vector idea we're going to be able to generate this rotating vector using some ideas from complex arithmetic and Euler's formula I find these to be a really beautiful pattern and it emerges from such a simple idea as a rotating vector you