 I actually get it down. So let's do our 45, 45 90 degree triangle first, okay. Now I'm going to draw 45 degree triangle here. This has to be the same size, right? Well, it's too big. It's 45, 45. Okay, so all I did was to be as accurate as possible when I drew the line. I put little ticks, where is it? I put a little tick here and a little tick here that measures the distance here and all I did was flip it and put it up, right? Since I don't have my protractor, they're all in a way in boxes somewhere. So right now what we got is a special triangle. This is 45 degrees. I mean, this is 90 degrees. This is 45 degrees, which happens to be the same as pi over 4 radians, right? So I'm going to put pi over 4 up here. So right now we've got degrees mixed in with radians, which you really don't do, but we're doing it so we can take a look at ratios and just remind ourselves what 45 degrees is. Okay. Now, for our special triangles in mathematics, what we've talked about is that I've mentioned a couple of times that what we like to do is we like to simplify our calculations, right? That's the beauty of mathematics. Math allows us to take a look at any system and try to simplify it as best as we can, right? By combining, you know, merging variables or setting up values for variables or for things that stay constant to something simple that allows us to do the calculations as quickly as possible for us to be able to scale things, right? For us to be able to manipulate them as easily as possible, right? One of the things we did initially, obviously, was we took a circle and to simplify our calculations, we took a circle and we said we're going to create the ideal circle, we're going to create the unit circle. And what we're doing is we set the radius to be equal to one and that made a unit circle, right? Because one is easily scalable, right? Easy to work with. So one of the ways we did this, to simplify our calculations, to set something at a value that makes it easy for us to do our work, we took a circle, we set the radius as one and created a unit circle. The second time we did this to, you know, we set things in our system to make our calculations easy, we took, we introduced the concept of radians, right? So instead of working with degrees, what we did, we took the radius of the circle, superimposed it on the circumference of the circle and we said, if we travel the same distance along the circle on the outer edge on the circumference of the circle, the distance equivalent to the radius, we call that one radian, right? So that's the second time we standardize something to make our calculations easy. This is going to be the third time. Now for us to figure out what sine, cosine and tangent are, we're going to standardize or use our special triangle. And the way triangles work, and we talked about this a fair bit. Now, if you recall from series one, we did talk about triangles a little bit, but just to recap, the way triangles work is just imagine this being our triangle, right? Not writing on this one, just imagine this being our triangle, right? Now in a triangle, an angle controls this opposite side, right? So this angle controls this side, this angle controls this side, and this angle controls this side, right? If you want to visualize this, just imagine this is our triangle. Now I'm going to decrease this side. If I'm going to decrease this side, which angle is getting smaller? It's this guy right here, right? So I'm going to decrease this angle and this side gets changed. These sides don't change, right? So an angle controls the opposite side, right? If I make it bigger, this angle gets bigger. So for us to do our calculations, what we end up doing, we standardize our special triangles. And the way we standardize it to make our calculations easy, we take the smallest angle, for this triangle, it happens to be 45 degrees, and across from the smallest angle, we set that side to be one unit, okay? So we're going to set this thing to be one, right? It's just like generating the unit circle, right? We set the radius to be one. It's just like us setting up radiant measures, right? One radiant is equivalent of us traveling the same distance as the radius around the circle. Mixer calculation is easy. This makes our calculations easy. Across from the smallest angle, we set that side equal to one. Well, that's 45 degrees. That's 45 degrees. Five over four radians is same thing as 45. If this is one, this has to be one as well, right? And if we use Pythagorean theorem, a squared plus b squared equals c squared, what we end up having is this guy becomes square root of two, okay? The distance here becomes the square root of two. Now, one could be anything you want, one meter, one meter. This is the square root of two meters. The square root of two is just the number, right? One point, whatever it is, okay? So to take a look at our trig ratios, to find out what our trig ratios are for this special triangle, we just go for sine theta. It's offset over hypotenuse. Cos theta is adjacent over hypotenuse. Tan theta is opposite over adjacent. And that's what our trig ratios are, right? So sine theta, and theta happens to be 45 degrees, or pi over four radians. So sine of 45, I should put down 45 here actually. So sine of 45 degrees, right? Sine of 45 degrees is one over root two. Now, one over the square root of two, if you recall from series two, we talked a lot about exponents and radicals, right? And radicals are something you got to have a really good grasp on. But there's a rule where it says we can't have an irrational number in the denominator. So what we end up doing is multiplying this by root two over root two, which happens to be one, right? One appears again, because you can multiply any number by one without changing its value, right? So what we end up doing is we multiply this by root two over root two, and that rationalizes the denominator. So this also appears as root two over two, and this is 45. So that sine of 45 degrees, cos of 45 degrees, I'm just going to write this as cos of pi over four, right? Because we're also working in radians. So cos of pi over four is adjacent over hypotenuse. So it happens to be one over root two, or if you're doing it here, one over root two. So this is also one over square root of two, which happens to be root two over two, right? Now you're going to see both of these appearing, depending on who your teacher is, what books you're reading. They use both. For me, they're both equivalent. It doesn't matter for representation-wise presenting the information. The answer is you put one over root two, or if you put root two over two. But if you're going to end up using these values, you want to work with this root two over two, because it makes our calculations much easier. Okay. So that's what cos of pi over four is, root two over two, or one over root two. Ten of 45 degrees happens to be opposite over adjacent, which is one over one. Okay. So ten of 45 degrees is one over one, which is just one. Okay. That's what our trick ratios are for the special triangle, for the special triangle. And if you recall what these values represent for a unit circle anyway, is our coordinate system, right? Sine theta for a unit circle is y, and cos theta is the x-value. So this coordinate here, when we're standing here, sine theta is our y-coordinate. So sine of 45 degrees is root two over two, right? So our y-part here becomes root two over two. And our cos is root two over two as well. And that happens to be our x, root two over two. So what this says is, when you go up from here to 45 degrees, your y-value, if we take this across, your y-value here is root two over two, right? That's how far you are up on the y-axis. And your x-value here is root two over two as well. Okay. That's the distance that you've traveled along the circle, along the x-axis. And that's the distance you've traveled up the y-axis, right? To get to this point 45 degrees on a unit circle. Pretty interesting, pretty cool actually. Let's do the other special triangle, 30, 60, 90, right? And this one we're going to draw approximately. So what we're going to do if we draw it here, we'll do our calculations here. Cool. So I'm just going to draw this guy first. So let's assume this is an accurate 30, 60, 90 degree triangle. Okay. So this is going to be 90 degrees. This is going to be 30 degrees. And this is going to be 60 degrees. And 30 degrees is pi over 6. And 60 degrees is pi over 3. And again, what we're going to do, we're going to standardize this special triangle, right? And the way we're going to do it, we're going to try to make it as easy as possible on our calculations across from the smallest angle, because this angle controls this side, the opposite side, across from the smallest angle, we're going to set that length there to be one. And what happens, the distances here, what the end up being is across from the 60 degrees ends up being the square root of 3. Okay. And across from the 90 degrees, the hypotenuse ends up being 2. And again, what we're going to do, we're going to take a look at where trig ratios are for 30 degrees and 60 degrees. So we're actually going to generate two of these, right? Because we're going to have to make one for 30 degrees, sine, cosine, and tangent. And we're going to have to make one for 60 degrees, sine, cosine, and tangent. So let's do these down here. So sine of 30 degrees is going to be opposite over hypotenuse. So that's 1 over 2. Cos of 30 degrees or cos of pi over 6 is root 3 over 2. Cos of over 6 is root 3 over 2. Okay, hopefully this is all going to come out. Are we too far down? Well, we'll squeeze it in. So tan of 30 degrees is 1 over root 3. So tan 30 degrees is 1 over root 3. I believe so, right on the edge. So tan of 30 degrees is 1 over root 3. And again, this thing, we have an irrational number in the denominator, right? And we can't have any irrational number in the denominator. So we multiply 1 over root 3 by root 3 over root 3, because we can multiply anything by 1 without changing this value, right? So all we're doing is we're going to represent this as another form. The value doesn't change. We're just going to make it look different, right? That's all we did here. We multiplied 1 over root 2 by root 2 over root 2. So we multiplied this thing by 1. So we didn't change this value. All we did, we changed this look, right? We made it look like something else. And this is the same thing. We're going to multiply 1 over root 3 by root 3 over root 3. So this ends up being root 3 over 3. That's what tan of 30 degrees is, right? And what these values represent is, if we go on the unit circle, if we go here, if we're standing at 30 degrees here, right? And remember, this is our reference angle. The angle in standard position would be 150 degrees, right? But what we're going to end up doing is working with reference angle a lot. This is, that's one thing we really need to do to analyze our movement around the circle, right? So if we do 30 degrees, sine theta is our y-coordinate, right? The y part of this. So if we draw our coordinate system here, our y here, if we come across, that ends up being sine of 30 is 1 over 2. So that's how far up we are on the y-axis. So our y part is 1 over 2, right? And our x part, which is cos theta, is root 3 over 2. That's how far down the x-axis we are. So this is root 3 over 2, right? Root 3 over 2. But since we're on the negative x coordinate, right? Negative part of the x-axis, this is negative root 3 over 2, negative root 3 over 2, right? And that's our coordinate, if we're standing right here, right? Based on our Cartesian coordinate system for a unit circle. Okay. Now let's take a look at what our sine cosine and tangent are for 60 degrees. So sine of, let's move this up a little bit, these things, right? So sine of 60 degrees is opposite over hypotenuse, so root 3 over 2. Cos of 60 degrees is 1 over 2. Cos of pi over 3, let's say. So far between degrees and radius, we're familiar with both. Cos of pi over 3 is going to be 1 over 2. And tan of 60 degrees is going to be opposite over adjacent, which is just root 3 over 1, which is just root 3. So tan of 60 degrees is root 3. You know, again, if we want to understand what these values mean, what sine theta and cos theta mean, what sine of 60 and cos of 60, cos of pi over 3 mean, they're just the coordinate systems of where we are here, right? If we're standing here, our y value is root 3 over 2. That's how far up along the circle we are. So this is root 3 over 2. And our x value, if we bring it down here, is 1 over 2, but negative because we're in a negative x-coordinate. So this is negative. So sine and cos of an angle actually give us the coordinates of where we are on the circle. They give us, you know, we've talked about this when we're graphing the sine function and cos function and tan function. And we're going to do this again. We're going to graph these trig identities, sine, cosine, and tangent based on radians, because we're going to work in radians a lot. But what they are is our y-coordinate and for sine anyway, and our x-coordinate, right? And that's what these numbers represent. They're just values, right? So these are our two special triangles we have to know. I'm going to refer to these a lot, right? Now, if you were paying attention, there's one thing that's happening here that you should have noticed, right? If this is our unit circle, that means the radius is 1, right? Now, I didn't put that unit circle here because I didn't want to confuse matters, right? This radius would have been 1, which means the hypotenuse would have been 1. But if you noticed, our special triangle, the hypotenuse, is not 1, it's 2. So what we're going to do right now, just to convince ourselves that these ratios don't change no matter how big the triangle gets or how small the triangle gets, the values don't change. What we're going to do is go back to something that a lot of people cover in my area anyway, in grade 8 or grade 9, where we're talking about similar triangles. So we're generating similar triangles just to convince ourselves that these values don't change for sine and cos. Basically, what we're saying is if you have similar triangles, and similar triangles basically means if you have triangles that keep the same form, but they're different sizes, right? One is a bigger version of another one, right? Like a mini, mini special triangle relative to a bigger special triangle, right? So what we're going to do is convince ourselves that the ratios of the sides don't change as long as the angles of the triangle are same.