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 ( S(\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, that yellow dot going up and down, that's the projection of the tip of the green arrow onto the Y-axis. 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. 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 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 ( S(\Omega t) ). Now if I go to a projection, this projection was onto the Y-axis. I can do the same animation, but this time project onto the x-axis, and that'll produce for us a cosine wave.

Let's see what that looks like now. Now in this case, if we switch over, you can see that the projection, that dotted green line, is onto the x-axis. What this is doing is it's producing a cosine wave.

So this is going to be ( \cos(\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°, 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.

Now I want to 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 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.