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# Sine and cosine from rotating vector

One way to generate sine and cosine is to track the motion of a rotating vector. Animated. Created by Willy McAllister.

## Want to join the conversation?

• So if y axis gives us a sine wave and x axis gives us a cosine wave as seen in , what would a z axis gives us?
• Here's an animation of what happens if you set the z-axis to represent the angle at different times. https://www.youtube.com/watch?v=r18Gi8lSkfM
• i dont know why, but the wavy graph in the video was lagging out my computer, so i turned it from 1080p to 144p and it was still lagging, why?
• So then just to verify, increasing the speed of rotation would increase the frequency / decrease wavelength, while increasing the circumference of the circle would increase amplitude?
(1 vote)
• Very nearly right. As rotation speed increases the frequency goes up and the period decreases. Increasing the circumference is an increase in amplitude, but I would call it an increase in radius.

Period is measured in seconds, wavelength is measured in meters. To know the wavelength you have to know what physical medium the sine wave is traveling through.
(1 vote)
• at , should it be -pi instead of pi
(1 vote)
• You might be right. The pi tick mark is pi seconds away from zero, in the past. The grid in this animation is a bit tricky since the axis is sitting still and the whole plot is evolving with time. If I put a minus sign in front of pi I don't know that it would make the graph clearer.
(1 vote)
• how do you make two parallel lines using this method?
(1 vote)

## Video transcript

- [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.