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

Sine and Cosine come from circles


3m read
·Nov 11, 2024

Now I'm going to clear off the screen here, and we're going to talk about the shape of the sign function. Let's do that. This is a plot of the sign function, where the angle Theta—this is the Theta axis in this plot—where Theta has been plotted out on a straight line instead of wrapped around this circle.

So if we draw a line on here, let's make this, uh, this circle a radius one. If I draw this line up here and it's on a in a circle, the definition of sine of theta—this will be Theta here—is opposite over hypotenuse. So this is the opposite side, and that distance is the opposite leg of that triangle is this value right here. So sine of theta is actually equal to Y over the hypotenuse, and the hypotenuse is one in all cases around this.

So if I plot this on a curve, this is an angle, and I basically go over here and plot it like that. And then as Theta swings around the circle, I'm going to plot the different values of Y. If it comes over this way, down here like this, right? You can see that that plots over there like that. Now when the angle gets back all the way to zero, of course the sine function comes all the way back to zero, and then it repeats again as our vector swings around the other way.

So sine of 2 pi is zero, just like the sine of zero. So every two pi, if I go off the screen, every two pi comes back and repeats to zero. Now I want to do the same thing with the cosine function that we did with sine, where we project the projection of this value onto this time the cosine curve down here. This has the cosine curve with time going down on the page, and our definition of cosine was adjacent over hypotenuse. The hypotenuse is one in our drawing.

So cosine of theta equals adjacent, which is X, the x value, divided by hypotenuse, which is one. So in this diagram, the cosine of theta is actually the x value, which is this x right here.

So let me clean this off for a second, and we'll start at the beginning. Let's start with the radius pointing straight sideways, and we know that cosine of theta equals zero is one. So if I drop that down, if I project that down onto the angle zero, that's this point right here on the curve.

Now as we roll forward, we go to a higher angle. This projection now moves to here on the curve. When the arrow is straight up, we are at this point right here. We go back through the axis. If we continue on, this projects down here, and we're moving this radius vector around in a circle like this. Eventually, this one will be at the same point as before, as the one above, but it'll be on this part of the curve here.

And when we get back to zero again, the projection is to this point here. So that's a way to visualize the cosine curve getting generated by a vector rotating around this circle. The cosine comes out the bottom because it's the projection on the x-axis. When we did the sine, it was the projection on the y-axis that produced the sine wave when we went this way.

So I like to visualize this because this rotating vector is a really simple and powerful idea, and we can see how it actually generates—a way to generate sine and cosine waves. You can see how sort of naturally they come out at different phases, right? The sine starts at zero, and the cosine starts at one. With this way of drawing it, you can see why that happens.

So this relationship between circles and rotating vectors and sines and cosines is a very powerful idea, and we're really going to take advantage of this.

More Articles

View All
The Moment That Broke His Memory | The Long Road Home 360
[Music] I don’t think I’ve been just Carl since that day. PTSD to me is not a disorder; that is a reasonable reaction to something traumatic that you have been through. [Music] Looking back, we were also green; we had no idea what we were doing. SolarC…
POLAR OBSESSION 360 | National Geographic
Eleven years ago was my first trip to Antarctica. I came down here to do a story about the behavior of the leopard seal. My name is Paul Nicklin; it’s my job as a photojournalist to capture the importance and the fragility of this place and bring this bac…
When You Stop Being Available, Everything Changes - Carl Jung
Have you ever noticed how some people seem to have an almost supernatural control over the environment around them without saying a word? They don’t shout. They don’t beg. They simply withdraw. And suddenly everything changes. The energy shifts. People st…
For Children With Clubfoot, Treatment Can Be Life Changing | Short Film Showcase
Now, nom Santos Kumara my Sonia Gandhi camera down. Watchmen Sergio de Rarawa the Armada over bodega. Rebecca’s terrific asana trauma para me repair yerba. Mr. Murata or Tamia Scylla’s NYADA vada pav watch for do far para trás. Elohim as additional Shahad…
Ben Silbermann at Startup School 2012
Well, first thanks a lot for having me. Um, it’s really exciting for me to be here in front of like so many people that all want to build cool things. I was getting ready for the talk last night, and I was going back through old emails because sometimes …
Lecture 11 - Hiring and Culture, Part 2 (Patrick and John Collison, Ben Silbermann)
Part two of culture and team, and we have Ben Silberman, the founder of Pinterest, and John and Patrick Collison, the founders of Stripe. Um, founders that have obviously sort of some of the best in the world at thinking about culture and how they build t…