Hello. Today we're going to learn about special right triangles in the unit circle. First thing I want to do is point out these are right triangles. So these this does not apply to other kinds of angles. So once you know something is a right triangle, you can learn a lot from it. For example, you can use the Pythagorean theorem and we're going to use that. Okay. So let me mark that as a right triangle. First kind of special right triangle to look at. There are only two um and the first one is a an isosesles right triangle. So if those two angles are the same and it's a right triangle, you know they have to be 45° each. Uh which means that the two sides also have to be the same. So if we call that bottom side X, then that means the left side has to be X, right? So if the bottom is five, the left has to be five. The bottom is 20, the left has to be 20. Um that's the nice thing about isocesles triangles. We can find out the third side by using the Pythagorean theorem. Okay, so if we have a^2 + b^2= c^2, then we plug in x's for a and b and we get that c is x * the of two. In other words, all you have to do if you know one of the sides is multiply by the of two and you get the third side. So we'll do some examples in a little bit about that, but if you know that the one leg is five and that means the other leg is five and the hypotenuse has to be 5 2, pretty straightforward. Okay, so those are called um uh 45 45 90 triangles. So yeah, if you're multiplying by 2 to get from the leg to the hypotenuse, I want you to think for a second. What would you do if you were going the other way? Well, the answer is that you would have to divide by two by by root two, sorry. Um and like I said, those are 45 4590 triangles. Pretty straightforward. Okay, the other kind of triangle is a 30 60 90 triangle. So, let me label those. Um, it's always good to kind of draw up a scale if possible and to mark remember that the smaller one's going to be 30, which makes sense. Um, let's mark that bottom leg x. And what we're trying to do is we're trying to find what's the other leg and what's the hypotenuse. Uh, and in order to do that, what I want you to do is imagine that we're going to flip the triangle that we have over on the other side like that over the long leg. Um, just reflect it. So what you get is you get a 60° angle down there. Well, if you look at it closely, you have a 60° angle in the bottom left, on the bottom right. That means that this top part also has to be a 60° angle. And we have an isos uh sorry, a uh an equilateral triangle, which means all three angles are the same. So all three sides must be the same. Okay. Um so that actually helps a lot. For one thing, that that that one down there on the bottom right is also x because we reflected it, which means the whole side is 2x. And if it's an equilateral, that means this side is also 2x. Um, that's very nice. So, in other words, if you know the short leg of a 30 60 90 triangle, just by drawing this little picture and thinking about it for a second, you can figure out, hey, that's got to be twice it. So, if that short leg was five, then the hypotenuse would be 10. Pretty nifty. Um, moving along though, if we want to find the other leg, it's a little trickier. We need to use that Pythagorean theorem again. So, let's label them A, B, and C. Uh, I'm going to fill in X for A and 2X for C. And if you do, uh, figure out the math, you can follow along. Right there, you get that B is X * 3. So, another way of thinking about this is again, if you have X, all you have to do is multiply by two to get to the hypotenuse and multiply by roo3 to get to the other leg. So let's look at that here in a few practice problems. Um the first kind I want to do is this 45 45 90 triangle. They gave us the hypotenuse as 26. So remember when going from the hypotenuse to the leg, the legs are always going to be smaller. So you want to divide by 26. Oh, I know this is a 45 4590 triangle because E and F those sides are marked with that little hash mark. That means they're the same length. And so they are uh they are it's a scaling if they're the same and or sorry it's an isosesles if if it's the same. So that means it has to be a 45 4590 triangle. Um so let's take 26 and divide it by 2. What we get is something with a radical on the bottom. We don't like that in math because it can look very different and still be the same number. So let's get that radical out of the bottom. You simplify it and you get 13 2 which is actually a lot prettier than 26 / the of 2. Um, now the other nice thing about a 45 45 90 triangle is that those two legs are the same. So we get that f is also 13 2. Um, and yes, in these kinds of problems, I'm going to give you one side and you just have to find the other two. All right. Uh, so I'll leave the uh bottom left down here. I'll leave that for you to do on your own the practice. That's another 4590 triangle. Let's look on the right side here. Here we have a 30 60 90 triangle. Uh, again remember if you have the short leg, which is that's the one we have. it's opposite the the angle 30° and it's right next to or adjacent the 60° angle, then all you have to do is multiply by two to get to the hypotenuse. So 6 * 2 is 12. Uh we could use the Pythagorean theorem for the third side. Or we could just remember, hey, uh remember the short leg times roo3 gives us the longer leg. So the answer is going to be 6 3 for y. Um, those are pretty straightforward and there's an easy way to check your work here. Let's just make sure we got that correct. It's simply using the Pythagorean theorem. Um, a^2 plus b²= c^2. So, let's plug those in the values we just got. We started with a um, and we plug in b and then c. And you can even do this. Need a little more space. You get 36 + 108 is 144. And that is correct. That checks out. So you can remember it's easy as ABC or well uh a^2 + b^2= c^2. Okay, that reminds me I really need my uh joke. Uh just know that you should never argue with a special uh special triangle because they're always right. Okay, moving along. I want you to try those two triangles for uh before coming to class. I'd recommend right now. They should be pretty straightforward. And uh yeah, do those and we'll check them in class. Um okay, so this next uh this next section is the unit circle. You might have also heard it referred to as the death star or a bunch of things you going to have to memorize. Um it is not as bad as it first sounds like. Um as long as we think about it and understand it. Okay, so that's that's my goal for you here. First thing I want you to draw your attention to is it's the unit circle. Why do we call it a unit? Well, a unit means one. So, it simply means the radius is one. But that also means that this line, this line, this line are all one. Okay? And that's important to remember when you're looking at some triangles in just a little bit. All right? So, just remember all those lines, they're all one. Okay? I hope you didn't write that down on your paper. So, um, yeah, unit circle. Another thing that's good to know is that's a right angle there. And, uh, yeah, I want to look at some of these other lines. why we drew these specific lines. That first one I went to, that's half of 90. So that's going to be our 45 degrees angle. And the other two lines are simply a third the way and 2/3 the way to 90°. So they are 60° and 30° as the red and the blue respectively. Um so uh let's go from here. Let's draw a triangle in on there. And if you don't want to draw it over your pretty 30°, um you can copy it over to the right like I do over here. All right. That triangle is actually one of the special right triangles we just learned about. It is in fact a 45 4590 triangle. Now the thing to remember like I said was that line in the middle that's one because that is the length of the radius on the circle. Once you know that since it is a 45 4590 triangle you can go from the hypotenuse to the uh to the side and you get that it is get that it is 1 over 2. Okay. But again that's not uh simplified very pretty in math. So we want to multiply top and bottom by roo2. Um we do that and we get 2 over2. And remember because it's a 45 45 90 triangle both those sides are the same. 2 over2 and roo2 over2. So the next thing I want you to consider is what if this was actually a coordinate plane. Okay, that center would be 0 0 and uh you know our x is left and right and our y is up and down. Then that point this point right here that I'm marking, okay, with the orange that point, where would that point be? Well, it'd be to the right this far. Okay, that's the length of the leg there, right? which we just found out was 2 over2 and it would be up this far. Okay, which we also just found out was 2 over2. So the exact location of that point is 2 over2 comma 2 over2. That's very nifty. Um and we found out that just by knowing special right triangles. It's actually the point on the unit circle at 45°. Let's uh sketch another triangle over here to the left to find another one of these points. As you can imagine, this point that we're going to find is this right here. Okay, that's actually the goal of drawing this triangle. But let's draw the triangle and see what happens. So, I'm going to redraw it over here if you don't want to mess up your pretty circle. Um, these angles, as you can picture, are that's the hypotenuse is one. Okay, this uh is 30° right here and that's 60° right there. the uh so yeah to get from the hypotenuse to the side on a 30 60 90 triangle we divide by two remember that's going from the large side the hypotenuse or sorry the hypotenuse to the smaller leg is um yeah divide by two so that's going to be 1/2 and to get to the other leg we multiply by roo3 from the short leg which would be roo3 * a half also known as roo3 over2 so just by measuring those two sides we find out that this this point right here because this goes to the left right here and its length is 3 /2 then this point has an x value of negative3 over2 and a y value of positive 1/2. Okay, the negative comes from because we're going left on the x axis and the positive is because we're going up on the y. All right, here I'm going to draw one more triangle. Let you figure out the sides. Uh, well, I'll give you the sides. I'll let you figure out how I got that. Notice that they're both negative because we're going down on the y, that's negative, and left on the x, that's negative. And we found out that that uh that point is just - 1/2, comma,3 /2. Just using those two special right triangles, we can actually find every single point on this unit circle. And that is what I want you to do in this next section. Okay? Okay, so if we look here, I'll go ahead and fill in that first point we found was 2 over2, 2 over2. The angle was 45° and you guys just learned about how to convert to radians, and that's going to be helpful very soon. So I'm going to go ahead and have you fill in all the radians as well around the unit circle. Um, so if that's 45°, that's pi over4 right there. Um, another thing that's easy is to notice that this point right here, if we were plotting it, would be right one because it's a unit circle and up and down zero. So the point itself is 1 comma 0. Easy to fill in for 0 degrees and 0 radians. Um I'll fill in give you a few others. The other one we found was negative3 over2 comma 1 and that was at 150°. And uh another thing I want you to notice this other uh right here little filler bit uh can confuse some students. It's just that they're having you fill in that that is 2 pi and 360° which is the same as 0 and 0 degrees. going one rotation around the circle. So, the rest of these I want you to fill in before coming to class. All right? If you're getting stuck, simply draw a triangle. And um yeah, if you can't if you're having trouble with this and just really don't understand what's going on, I want you to draw a triangle for each one of these on a separate sheet of paper. That way, you have something to bring to your classmates when you're working on this. Hope you enjoyed this video and hope you learned something. Thanks.