The last thing I want to talk about today is another powerful mathematical function that we can use a lot enjoying, and, the, these functions are, trigonometry which he, probably studied in high school or the equivalent. and they are three functions that relate to triangles. relate angles and triangles. So, if we look at the slide here, we've got a right angled triangle. We've got an angle A in that triangle, and then we can label the sides. The adjacent side, which means the side that's next to the angle. The opposite side, ones opposite it. And the hypotenuse, which is the long the long angle, the long side of the triangle. And this trigonometric functions are defined in terms of these three sides. The sine of an angle is the opposite side to that angle divided by this hypotenuse, the length of the opposite side divided by the hypotenuse. In the case of the tangent, it's also defined the same way. Now, we're not actually going to use these to work with angles or triangles, and I'm going to show you something different. I'm actually just interested in working with sine as a function, because it's a really instant function. And it's really powerful for doing graphics animation, and I'm going to show you what it does. So, if we're drawing the circle, you can that the vertical line in the circle is the opposite of the angle in the middle of the circle as we rotate around. So, we can see we're drawing a circle, and I'm tracing out the path of the actual sine function at the same time as drawing the circle. And you can see, is that when the angle is. When it starts again, it's near zero, the sine function zero. It gets increasingly more negative, then increasingly more positive. And then it goes back down to zero again, and it starts the way around a circle. So, it repeated itself exactly, going negative, going positive again. And what's powerful about is that it's repetitive oscillating function. So, it goes up and down, up and down, up and down all the time. It looks a little bit like a wave. In fact, it's a very important mechanism for modeling sound wave. Sound waves act a lot like sou, sine waves. In fact, we can build up any sound wave out of a whole load of si, of si, of sine functions of different sizes and frequencies. So, what do we, what can we do with this? Well, before we come to that, I'll talk a little bit more about the function. So, this is More detail about the sine, sine function, this is how we would write it out. we're using the function sine, which is the calculates the sine of an angle, but we're putting some other numbers in. Firstly t, we can think of this as time, if you're working with animation with sounds, the sine function is changing over time. Then we've got, we multiply the result by A. That's called the amplitude, that's how high it gets and also how low it gets. That's how big the sine wave is. In audio, that is closely related to the volume of audio. If you take, if you're listening to sound. Then we have the frequency, that's how many of the ups and downs you get in a second, say, if you're dealing with time. and that what I'm showing you in the diagram here is that the distance from one point to next identical point is one divided by the frequency of wave. And finally you got the phase, and that, we can really just think of that wave's as a starting point to the sine wave, say how off, far off the zero point your actual function is starting. And you can combine together at different sine waves at slightly different phases, and you get an interesting effect. Say, you're starting one at point, and you're starting another one a little bit later, and you combine them together, and that'll [INAUDIBLE] an interesting visual [INAUDIBLE] effect. And there's a lot you can with sine waves. So, a sine wave can be used nicely in graphics. It has a nice shape. it starts to look a bit like a water wave. You can [INAUDIBLE], you could model water with sine waves. And certainly, if you add a bunch of sine waves together, you can get quite a nice model of it, or even landscapes. And importantly you can use in animation. And that's really powerful because the great thing about sine wave is that it oscillates. It goes up and down repeatedly. And that's a really useful thing for animating moving objects. And we'll see make work, work a lot with sine wave [INAUDIBLE]. The other great thing for sine wave is, as I said, it's a fantastic tool for modeling audio functions, and I think Matthew will talk about that in a later [MUSIC] [MUSIC]