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## AC circuit analysis

# Sine and cosine from rotating vector

## 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.