 In this video, suppose we want to solve a story problem involving the law of science, which the law of science will come up in just a second. So we're living on our good blue marble planet Earth right here, which looks from the side will look like a circle. And we have a satellite that is rotating around planet Earth. Okay. And so let's take two points on the surface of the Earth for the sake of it. Let's call these points B and D, and we can measure the distance between them. So along the crust of the Earth, this will be 910 miles between these two cities. Okay. And then the angle at this given moment of time. So let's say that the satellite is directly above point B right here. Right. Let's say that the angle that the satellite forms between these two cities B and D. Let's say that the angle is 75.4 degrees like so. So we know that the radius of the Earth is 3960 miles. And so that tells us that the length C B and the length CD would both be about 4000 miles right there. So using that information, can we find out how far above the Earth is the satellite? So how high up in the altitude is it? So let's call that distance X. We want to solve for X according to this diagram right here. That is our, that is our task to do. And that would give us the height, of course, of this satellite here. So notice that this distance right here, the distance between B and D, this is not a straight line. This is the arc of a circle. And so we know that the arc of this, the arc of BD is given, of course, as 910 miles. Right. That's important because if we think of the circumference here of a circle, we can get back to the arc length formula we saw before, right? The arc length S is equal to the radius times the angle if that angle, of course, isn't radians. And particularly you can solve for the angle right here and we get that theta is equal to arc length over R. So we can measure this angle right here, theta, if we so chose to, we could measure that using this arc length formula, right? So that's going to give us the measure of angle C right here. It's going to equal the 910 miles, which was the arc length, divided by the radius of the circle, which is 3960, the radius of the Earth right there. And this is going to be a measurement in radians, of course, right? This is not in degrees. I need to make mention of that. We can simplify this thing. I mean, clearly 910 and 3960 are both divisible by 10. So let's do simplify that a little bit. You get 91 over 396. Does that fraction simplify anymore? I'm not sure I care at this moment. Because the angle of the measure of angle A is in degrees, we have to convert least, we have to either convert angle A into radians, we have to convert angle C into degrees. We have to go one way or the other. Let's just convert angle C into degrees. To convert from radians to degrees, we're going to times this radiant measure by 180 degrees divided by pi, like so. And there are some common factors that come into play right now. We can simplify this thing. Again, I'm not going to go through all the details of the arithmetic here, but 180 and 396 do have some common factors. So this would simplify to be 91 times 5 degrees over 11 pi, like so. And we can take 91 times 5, that gives us 455 degrees over 11 pi. And we're going to leave this exact right now. I mean, I can approximate if you want it to. I mean, I guess it's not too harmful to do that. And so we'd get that the angle theta angle C is approximately 13.2 degrees like so. And so yeah, yeah, I actually do change my mind. I do want to use the approximation. There is some a bit of a rounded error here, but our goal is actually to look for the measure of angle D at this moment. What are we going to do with angle D? Well, if you add up the three angles of the triangle A, C, or what is that, A, C, D right there, the measure of angle D plus the measure of angle C plus the measure of angle A, this adds up to be 180 degrees, like so. And in which case, then we can solve for the measure of angle D very quickly. We get 180 degrees. Take away 75.4 degrees, which we can assume that's an accurate measurement. We don't know where that came from. You have to subtract the measurement of angle C, which is 13.2, which that is rounded. So it's not perfect, right? We could use the perfect value we wanted to. But the error of C will be the same at the error of D. So D will not be more aromies than the rounding error that happened with C. And so do make this subtraction here. You're going to get, if we take away 75.4 and 13.2 from 180 degrees, feel free to use a calculator to help you with the decimal arithmetic here. We get the measure of angle D, not B. The measure of angle D there is going to equal 91.4 degrees, like so. So I'm going to redraw our triangle for a moment so you can see where we are in the process. All right, so we have this triangle. Looks something like this. And so this was D. And we get that the measure of D turned out to be, remember, 91.4 degrees. We have the measure of C right here, which turned out by our approximation to 13.2 degrees. And then there was angle A over here, which measure was given to us as 75.4 degrees. Now I should mention that this diagram is not drawn to scale. That is perfectly fine. We don't need to worry about that. This is just about spatial recognizing things, not about actual drawing things to scale. So what else do we know about this? This side length was 39.60. This one, there's some point in the middle B. So there's some unknown distance X, which we want to know, but there's also this distance 39.60, like so. And so look what we have set up right here. You have angle D and we have this side length over here. That forms an AOS, an angle opposite side. It has an unknown and an X, but there's also another angle opposite side we know with associated to angle A. We have angle A in its opposite side, which is the radius of the earth here. Using the law of signs, we can put this stuff together. So if we take sine of capital A over little A, this will equal sine of capital D over little D, like so, for which sine of A, we have here this 75.4 degrees. This is over little A, which is going to be 3960. Then we get sine of D, which is 91.4 degrees over little D, which is 3960 plus X. We don't know what X is, but we have to solve for X here. So we're going to cross multiply here to solve for X, cross multiply like so. So we end up with 3960 plus X times that by sine of 75.4 degrees. This equals 3960 sine of 91.4 degrees. We need to solve for X here. So we're going to distribute sine of 75.4 throughout on the left hand side. This then gives us X times sine of 75.4 degrees. Then we're going to get plus 3960 times sine of 75.4 degrees. So I'm going to delay approximation till the very end of this exercise here, again, to avoid rounding errors, which we don't want to have if possible. So we get three, on the right hand side, we get the 3960 sine of 91.4 degrees. We want to move this friend to the other side of the equation. We can do that by subtraction, of course. And so we then get X times sine of 75.4 degrees. This is equal to, well, on the right hand side, everything's visible by the 3960. So I'm actually going to factor that 3960. And then we get sine of 91.4 degrees minus sine of 75.4 degrees. Although there might, although there may be temptation there, just because we have a sine, not as a sine doesn't mean we can combine them together anyway. That's not how angle differences work with sine. We can't just subtract the angles. We can just leave it alone. And then to finish off, we need to solve for X, which the coefficient of X right here is sine of 75.4 degrees. That's just a number. If you had something like 3X was the left hand side, you divide both sides by 3 and go on from there. The fact that our coefficient is some irrational sine of an angle is no big deal. We'll divide both sides by sine of 75.4 degrees. Do it to the right hand side as well. We get sine of 75.4 degrees. These signs cancel out and we've now solved for X. X then equals 3960 times sine of 91.4 minus sine of 75.4 degrees. And this all sits above sine of 75.4 degrees. Which you can't cancel these ones right here. You'd have to cancel it from the, you have to cancel it from the whole thing, which is unavailable here. So let's just leave it the way it is. This is perfectly good. If you wanted to do some type of cancellation, again, you could break up the numerarity if it wasn't factored, but you'd still have something that's basically as complicated as you see in front of you. So you could end up with 3960 sine of 91.4 degrees over sine of 75.4 degrees minus 3960. That's how the fraction would simplify if you wanted to. And I mean, there's one less sign in play here, but it's really not much of an improvement, whichever way you're going. But whatever, take one of these forms, plug it into your calculator. Make sure you plug it in correctly. Follow the order of operations for your calculator. The syntax is a big deal. Make sure it's in degree mode. And when you use your calculator, this would approximately give you 130 miles. Distance here is measured in miles here. And therefore, that would be the approximate height of the satellite above the earth as it's spinning around. And so in this video, we then demonstrate how we can solve this basic word problem, a story problem using the law of signs. And that's what we've been focusing on, of course, in lecture 24. And if you learned anything about the law of signs, I hope you did. Give this video and any of other videos alike. Subscribe to the channel if you want to see more videos, math videos like this in the future. And if you have any questions, feel free to post them in the comments below and I'll answer them at my soonest convenience. Bye, everyone.