 In addition to the 30-60-90 triangle, another very important type of right triangle is the so-called 45-45-90 triangle. So as the name suggests, it'll be a right triangle, so it does have a 90-degree measurement. The other two are gonna be both 45 degrees. So this is a special case because this is the situation for which the right triangle is likewise an isosceles triangle. And so an example of an isosceles right triangle can be found right here. So we see that the angle measures are 45, 45 degrees. Of course, it's a right angle as well. If the legs are side length one and one, then the hypotenuse will be the square root of two. And therefore, by the usual ratio, if you're looking at sine here, sine of 45 degrees, the sine is gonna be opposite over hypotenuse. So sine of 45 degrees would be one over the square root of two. Now, if you're used to rationalizing the denominator, you can times top and bottom by the square root of two and then you end up with this figure right here, the square root of two over two. Notice the square root of two times itself gives you back a two. So sine of 45 degrees is the square root of two over two. Likewise, if you cosine, you'll take adjacent over hypotenuse and you'll again get one over the square root of two, which is equivalent to the square root of two divided by two right there. And so that takes care of sine and cosine of 45 degrees. But how did we know these were the side lengths that work out for a 45, 45, 90-degree triangle? Well, the idea again is that it's an esosceles right triangle. So if we have our right triangle, for which it's a esosceles, what that means, don't worry about my diagram here being perfect. If it's an esosceles triangle, the two non-right angles have to be the same. So how do we know they're 45 degrees? Well, if you have an angle, an angle plus 90 degrees, this adds up to be 180 degrees. Well, it's attracting 90 from both sides. You get that two X equals 90 degrees, divided by two, you see X equals 45 degrees. So if it's an esosceles right triangle, the two non-right angles have to be 45 degrees. Okay, what about the other sides? Well, just for the sake of argument, let's say that one of the legs is one. Well, because it's an esosceles triangle, the two legs will have the same size. They're congruent to each other. So this side length has to be one as well. Well, what about the hypotenuse? We don't know what it is. If you call the sides ABC as usual, let's call the hypotenuse C for a moment. We get one squared plus one squared equals C squared. So you're gonna get one plus one equals C squared. That is C squared equals two. Taking the square, you see that C equals the square of two. And so that then justifies the picture we have right here. Every esosceles right triangle will be proportional to this one right here. Thus giving us these very simple trigonometric ratios. Well, let's remove these specific numbers for a moment. And let's suppose that the leg of this esosceles right triangle is arbitrary. Let's call it X for a moment. So we know it's a 45, 45, 90 degree triangle. And we have that one of the legs is X. Well, because it's an esosceles triangle, the other one has to be X. So whatever this one is, just copy it over here. That's all we have to do. And then to get the hypotenuse, you're just gonna take X times the square root of two. That's all it takes to get there. And then go in another direction. If we start off with the hypotenuse, let's say the hypotenuse is X. Well, then to get one of the legs, you're just gonna take X divided by the square root of two and then copy it for the other leg as well. So we can compute the side lengths of a esosceles right triangle very quickly. So for example, imagine that a 10 foot rope connects the top of a tent pole to the ground. If the rope, well, okay, let's just draw that picture for a moment. Let's, it's safe to assume that the tent post is going perpendicular with the ground right here. And so then the rope is doing something like this. And so we know that the rope is 10 feet long. That's what we know. All right, so then continuing on with the story here, if the rope makes an angle of 45 degrees with the ground, again, we're assuming this is a right angle. That means that the other angle is gonna be 45 degrees. This is a 45, 45, 90 triangle. That is an esosceles right triangle as stated right here. How tall would the tent pole have to be for this situation? Well, if we take the hypotenuse to get to the short side, you have to divide by the square root of two. If you take the leg and wanna go to the hypotenuse, you times it by the square root of two, but to go from the hypotenuse to the leg, you divide by the square root of two. So this side would have side length 10 divided by the square root of two, which is approximately 7.0711 feet. And so we could say with confidence that the tent pole is approximately seven feet long.