Sine and cosine from rotating vector

- [Voiceover] Now, I'd like to demonstrate one way to construct a sine wave. What we're gonna do is we're gonna 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 gonna 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 and 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 around and around, you could 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 gonna switch to a new animation and 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 'til 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 onto the x axis. And, that will produce for us a cosine wave. Let's see what that looks like. Now, in this case that we've switched over, you can see that the projection, that dotted green line, is onto the x axis and what this is doing is it's producing a cosine wave for us. So, this is gonna 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 hits 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 cosine wave. And, it has the same frequency as the sine wave we generated. And now I wanna show you these two together because it's just sort of a beautiful drawing. I'll leave our animation here for a second. We see our 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 gonna 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.