 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 Missildine. In this lecture 27, we're going to talk about finding the area of a triangle. We're used to the very famous formula of a triangle's area, that is, area is equal to one half base times height. But where does that formula come from, and how does actually one use that in a practical sense? It turns out that for a right triangle, it's pretty obvious to see why that works. Because for some right triangle, you have something like this, in which case, if the right angle is this one right here, the base is just say this side of the right triangle, the height would then be the other one. Notice in particular, this is just the legs of the right triangle. Then with this right triangle, you can think of it as half of a rectangle, kind of filling in the details right there, where you basically just have two congruent copies of the same triangle, for which then the rectangle, it'll have its length and its width being the height and base of the original triangle. But as the right triangle is half of the original area, you get the one half base times height. So that works great for right triangles. And oftentimes when a student's required to compute the area of a triangle, it nearly always is a right triangle. Or if not, an altitude of the triangle is pretty much explicitly given to them, so that they can take the base to be the side of a triangle, then it's corresponding altitude. But of course, what if the altitude's not there, right? Well, we'll get to that in just a second. Let me show you how it kind of works in this more general setting here. Because if you have something like this, where you have an oblique triangle, it's not a right triangle. Again, it's the same basic idea. You cut it along an altitude, you make two right triangles. And so you double this right triangle right here. And so here's a rectangle, something like this, whose height would be h and whose length would be, oh, I don't know, let's call this distance x. So this would then be c minus x. You get h times c minus x for the first rectangle. And then for the second rectangle, you double this right triangle right here, and then you'll get h times x right here. And that would then be the area of said triangle. When you add these together, since there's the common factor of h, you're gonna get h times c minus x plus x. The x's cancel out, you end up with h times c here. So again, the altitude times the base. This of course would give you the area of the rectangles. So we need to take half of that, of course. And so you end up with this one half, the height, the altitude there in the base c. And so it's the same basic idea. You can also make an argument that a triangle is half of a parallelogram and try to make a simpler statement there. But I think that rectangle approach is probably more familiar to us. So, okay, the area of a triangle is one half it's base times height where the height is the length of some altitude. But what if the altitude is not specifically given to us like it is in the case of right triangles, okay? Well, we actually wanna use some trigonometry to help us compute this altitude. And therefore this area of formula, one half base times height, you really should think of it as area equals one half this times the side length of a triangle times the length of the corresponding altitude. We can modify this formula using a little bit of trigonometry using the law of sines and law of cosines. In this video, we're gonna focus on the situation known as the side angle, side situation, where that is we know the side of a triangle, we know an angle of triangle and we know another side of the triangle. And in particular, the angle is gonna be the angle between these two sides, side angle side. If you know, so this side length and this side length and then the measurement of the angle between them, we actually are in a position where we can compute the area of this triangle. Because if you look at this, right, that you see the altitude right here, the area of this triangle is gonna equal like we saw before, one half base, which for us will choose the side length C times height, which in this case would be the altitude H. You get one half C H. Well, C is given to us, but what about H? If we focus on this right triangle here, just the left-hand side, we know the measurement of angle A. And so the altitude is the opposite side with respect to angle A. And then B would be the hypotenuse of that right triangle. Therefore, we get that the height, well, let's say it this way, we're gonna see that the sine of A is equal to H over B. If we times both sides of the equation by B so that they cancel on the left-hand side, you end up with H is equal to B sine of A. And so you put that in over here. You get one half C times B sine of A. If you wanna put the letters in alphabetical order, you end up with area is equal to one half B C sine of A like so where B and C are the sides of the triangle that are not opposite of angle A. And you get this nice little area formula which is equivalent to our one half base times height. It's just we use trigonometry to explicitly say what the height of the triangle is gonna be with respect to the base C. And so that gives us one version of this formula. Area equals one half base times height. So it'd be one half B C sine of A. And we constructed this using angle A and the altitude associated to C, all right? You could of course change your focus a little bit. If you knew the sides C and A and therefore the interior angle B, you could then construct an altitude here with respect to either angle C like the diagram shows or angle A would be fine. And you could show the area equals one half A C sine of B or similarly, if you knew the side lengths A and B and you knew the interior angle C, then we could see that the area is equal to one half A B sine of C. Similar arguments would apply there. And so notice what's going on here that you use all three pieces of the triangle. So you have your triangle A, B, C. So in either case, you choose an angle to play here. Angle A, angle B, angle C, maybe that's given to you. And then you're gonna choose the side lengths that are not opposite of that angle. So you have B, C, A, you have A, C, B, you have A, B, C. All three letters show up, one of them's an angle, the other two will be lengths. So let's compute the area of a triangle when we know the side angle side conditions. So let's say that we know side length B is 25 inches. Angle A is 60 degrees and side length C is equal to 32 degrees, sorry, 32 inches. So this is a side angle side situation that we have going on right here. So the area we saw previously, this one half base times height, this would equal one half, take the two sides we know. So B was 25, C was 32. In the middle, if you mix those two up, doesn't make much of a difference. And then we have to take sine of 60 degrees like so. 32 is an even number, it's divisible by two, so that's gonna give you 16. 16 times two is 32. 16 times 25. Sine of 60 degrees is actually root three over two. That's when we memorized there. Two goes into 16, of course, eight times. So you get eight times 25 times the square root of three. Notice that eight, of course, is four times two. Four times 25 is 100, so we're gonna end up with 200 times the square root of three. It's important to get the exact answer here because maybe that's what's required of us, but from a practical, since we probably need to approximate this as well. So 200 times the square root of three is approximately, we'll consult the calculator. If we round to the nearest square inch, we would end up with 346 square inches like so. And this example illustrates exactly how we can compute the area of a triangle. If we know the side angle, side condition, you'll use one of these formulas, area equals BC sine of A, area equals one-half AC sine of B, or one-half AB sine of C. You always have A, B, and C showing up here. One is the angle, the other two are the sides, and it's a very nice formula. You don't necessarily need to memorize all of these formulas. If you remember one-half base times height, and you remember where this formula came from, just recognize, and we just use the basic sine ratio to represent the altitude, which is unknown, using the side and angle that we do know, then you can reconstruct the formula. And so again, the memorization burden can be removed when we understand where this thing comes from. Believe it or not, when we understand the formulas, it makes them easier to memorize in the end.