yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Sine and cosine from rotating vector


2m read
·Nov 11, 2024

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.

More Articles

View All
Compliment/complement and desert/dessert | Frequently confused words | Usage | Grammar
Hello grammarians! Continuing our journey through the world of frequently confused words, I’d like to begin with “complement” with two e’s and “complement” with an i and an e. How the heck do we keep these separate? Well, first let’s get some definitions…
10 BAD@SS Online Games!
[Music] It’s fun to dress up your elf princess. But if you’re looking for an online game that’ll put some hair on your chest, let’s take a look at 10 incredibly badass online games. First, Mechanical Commando, a top-down shooter that asks, why would you …
The Stock Market JUST Went From BAD To WORSE | How To Prepare
All right, so this is getting out of control. First, we had Facebook plummet 26% in a single day. Then we had Netflix fall 35% overnight as they lost subscribers. Shortly after, Walmart drops 25% on missed earnings. And then the nail in the coffin: Tarjay…
How to Overcome Yourself | Nietzsche’s Superman
When you hear the word ‘Superman,’ you might think of Henry Cavill with superhuman abilities wearing a tight blue outfit and a red cape. But this superhero, originally from American comic books published by DC Comics, is not the original Superman. In fact…
How to Stop Procrastination Right Now | The 3-2-1 Rule
Hey, it’s Joey and welcome to Better Ideas. I was just sitting in my apartment and realized that I really needed to do my laundry. I’ve been putting it off for like the past two days or so. You know, I’m a busy guy, and every time I thought about doing my…
Saving One of the Most Pristine Wetlands on Earth | National Geographic
The Aqua Bengal Delta, one of the most vibrant wetlands on the planet, could be in danger of disappearing. From deep in the my humble forest highlands of Angola flow the surface water that feeds the Cuiúto and Cubango rivers. These vital waterways, flowin…