 Welcome back to our lecture series math 1060 trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture six, we're going to talk about the idea of reference angles. That is, how can we deal with the trigonometry of any angle under the sun? Well, it turns out we can always reference it to an angle in the first quadrant. In the first couple of videos for lecture six, we're going to focus on some special angles for which the trigonometry is much simple than the general case. In this video, we're going to talk about the so-called 3060 90 triangles. What does that mean? We're talking about right triangles, and so a right triangle always has a 90-degree angle measurement. A 3060 90 triangle, these are just the measurements of the angles in the triangle. A 3060 90 triangle is a right triangle with a angle measure of 30 degrees and an angle measure of 60 degrees. Notice that 30 and 60 degrees are complementary, and so that makes the appropriate angle sum for a right triangle. When it comes to a 3060 90 triangle, every 3060 90 triangle will be proportional to the one you see right here. That is to say this side opposite of the 30-degree angle, you could give us length of one. The side opposite the 60-degree angle, you could give the square root of three, and then the hypotenuse would have a size length of two there. Every 3060 90 triangle is proportional to this one. With this triangle, we can talk about the various trigonometric ratios. Let's think of sine for example, sine of 30 degrees. Looking at this angle measure right here, sine is opposite over hypotenuse. You give one over two, sine of 30 degrees is one-half. Cosine of 30 degrees would be adjacent over hypotenuse. You get the square root of three over two, which is cosine of 30 degrees. Similarly, if you change your focus to be on the 60-degree angle right here, we get that sine of 60 degrees, that is opposite over hypotenuse, root three over two. And then cosine of 60 degrees, you take adjacent over hypotenuse, one over two. So you get cosine of 60 degrees is one-half. So because of this triangle we have right here, we can determine the sine and cosine of 30 degrees and 60 degrees. But how do we get these numbers? What's so special about 3060 90 triangles? Why is this true? Well, it turns out that the 3060 90 triangle is really just a derivative of an equilateral triangle. That is to say, if we take an equilateral triangle, this is a triangle where all three sides have the same length. And for the sake of simplicity, we're gonna say the length of these sides are each two. So the side of our triangles are each two. Well, an equilateral triangle is likewise equilangular. That is to say, each of the angle measurements have the same angle measure. Well, if you have three angles of the same size, x plus x plus x, this equals 180 degrees, you're gonna get three x equals 180 degrees. In other words, x equals 60 degrees. So for a equilateral triangle, every angle measurement is necessarily 60 degrees. So you start to see how 60 degrees plays into this game here. Now, if we construct an altitude for an equilateral triangle, pick any of the vertices, so I'm just gonna pick the one on the top and we're gonna go to the bottom. Remember, an altitude is a line segment that connects the vertex, well, it connects one of the vertices of the triangle and forms a right angle with the opposite side, a so-called altitude. These are important for finding areas of triangle and such. Well, because a equilateral is perfectly symmetric as a triangle, it turns out that an altitude cuts an equilateral triangle in half, okay? And so what I mean by that is that the other side is gonna get cut in half. So the foot of the altitude here is gonna be the midpoint of this line segment. So this side is one, this side is one. It cuts it in half. But the altitude also bisects the angle on top. That is, it cuts the angle in half. Well, if you have a 60 degree angle on top and you cut it in half, you end up with a 30 degree angle right here. And so you can see that a 30, 60, 90 triangle is essentially just half of an equilateral triangle, okay? And so that's how we get this thing started. So you have a 30, 60, 90 triangle. The short side that's opposite of the 30 degree angles would be one because the original side length of the equilateral triangle is two. But this turns out to be the hypotenuse of it. So we have a triangle which has a leg equal to one. It has a hypotenuse equal to two. And then if you take the other unknown side, call it H for the height of the triangle, then we're gonna get that one squared plus H squared is equal to two squared by the Pythagorean equation. So we get one plus H squared equals four. Subtract one from both sides. You get H squared equals three. And therefore H equals the squared of three. Thus giving us the picture we started off with right here. Now another observation that's very important when you're working with 30, 60, 90 triangles is because of this picture we have right here and every 30, 60, 90 triangle, they're similar triangles, so they'll be proportionate. This triangle you see right here. An important observation to note here is that the smallest side, which was always opposite of the 30 degree angle is exactly half of the hypotenuse. That is, if you know the short side, then you can get the hypotenuse by doubling the length of the short side. Conversely, if you know the hypotenuse, you can get the short side of a 30, 60, 90 triangle by cutting the hypotenuse in half. Likewise, if you know the short side, the medium side will just be the short side times the square root of X. And conversely, if you know the medium side, then you can get the short side by dividing by the square root of three. So in summary here, if you have a 30, 60, 90 triangle, something like this. So again, we know that this is a 30, 60, 90 triangle. Then if you know the short side, the hypotenuse is always two X and the medium side will always X times the square root of three. This relationship simplifies calculations with 30, 60, 90 triangles. So for example, if you have a 30, 60, 90 triangle ADC and it's shortest side is length five, what can we say about that triangle? Let's just draw the picture real quick to help us out. So we know it's a right triangle. It's 30, 60, 90. So let's say this is the 30 degree angle. This is the 60 degree angle. And we know that the shortest side is length five. So again, that's gonna be, the shortest side is always the opposite of the 30 degree angle. In fact, the bigger the angle, the bigger the side. The smaller the side, the smaller the angle. There's this relationship there. Well, since it's a 30, 60, 90 triangle, we know the hypotenuse will be two times five. The hypotenuse will have length 10. What about the other side there? Well, since it's the side opposite the six degree angle, we take the short side five and we times that by the square root of three. And so we see that the side opposite of six degrees will have a length of five times square root of three. As another example, consider a ladder leaning against a wall. And this is a classic example we see in trigonometry. If you have a ladder leaning across a wall, something like this, then assuming the wall is perpendicular to ground. That's, unless you're in like the Tower of Pisa or something, that's a fairly safe assumption there. A ladder leaning against a wall forms a right triangle. Suppose that the top of the ladder is eight feet above the ground. So we have this distance right here is eight feet. And that the bottom, the angle with the bottom of the ladder with the ground forms a 60 degree angle. So we know this right here. Well, then this makes us a 30, 60, 90 triangle. And so if we wanted to figure out, for example, how long the ladder is, well, let's first figure out the short side, right? Since this is the side opposite of the 60 degree angle, to get to the short side, we have to divide by the square root of three. So we get the short side is gonna be eight divided by the square root of three. For which if you want an approximation, this would be 4.6188. If you're around to the nearest four decimal place there. And this is gonna be feet. So that just tells you that, okay, yeah, the ladder is about 4.6 feet away from the wall. But how long is the ladder? How that would be the hypotenuse right here. To get from the short side to the long side, or to the hypotenuse, I should say, we're gonna times that by two. So you get 16 over the square root of three, which again, using your calculator, gives you approximately 9.23376. So we could say that the ladder is approximately nine feet in length. And so these 30, 60, 90 triangles are special cases, but notice that we don't need to use the sign, cosine or tangent ratios to find the missing sides of a triangle. If it's 30, 60, 90 triangle, you can use this relationship about doubling or squirts of three to find the missing sides.