A surveyor wishes to find the distance across a river while standing on a small island . If she measures distances of a = 30 m a=30m a = 30 m to one shore, c = 60 m c=60m c = 60 m to the opposite shore, and an angle of B = 100 ° B=100\degree B = 100° between the two shores, find the distance between the two shores .

Hey, everyone. So in this practice problem, there's a surveyor standing on this little island inside of this river, and they're trying to gauge the distance between the two shores by using some trig. Basically, what they find is they measure the distance to one shore, which is 30 meters, the other, which is 60, and the angle between, which is a 100 degrees. We're supposed to find the distance between the two shores. What letter is that in this triangle? Well, if I got these things, laid out, I've got these letters over here. It's little c, little a, and big b. Then, basically, what I'm trying to find is this distance over here, which is gonna be the opposite of angle b. So this is gonna be the letter little b. That's what we're trying to find here. How do we solve this? Well, whenever we have a side and its corresponding angle, we can always use the law of sines, but we don't have that here. Notice how that we have one side, but not its angle, another side, but not its angle, and then one angle, but not its corresponding side. So whenever we have this situation, we have to use the law of cosines. Alright? So remember, there's 3 sort of variations of the law of cosines depending on which letter you're trying to solve for. Basically, what we're gonna do here is we're just gonna use the law of cosines to figure out what this b variable is by using the second sort of variation of the law of cosines. Alright? That's really all there is to it. So in other words, we're gonna find b by finding b squared, and, basically, this is just this equation over here. Remember that it kind of looks like the pythagorean theorem, a squared plus b squared equals c squared, but it's got an extra term over here. And the pattern is that it's always the other two variables, a and c, and then minus 2 times a and c times the cosine of the angle, which is big b. Okay? So if you look through the variables, you'll actually see that you have all of these letters, a and c and big b. So we can go ahead and just plug and chug. Okay? So this is just gonna be a squared, which is gonna be 30 squared, plus 60 squared, and then I'm going to minus 2 times 30 times 60 times the cosine of the angle between them, which is going to be a 100 degrees. Alright? So if you go ahead and actually work this out, what you're going to get, is you're going to get a really, really big number. And when you take the square roots of that number, you're just gonna get this is 71.6 meters. Alright? So that's a pretty sort of straightforward calculation over here. Just make sure when you plug this stuff in, you do it correctly. I like to sort of just do those two numbers, square them first, and then sort of subtract this really big multiplication over here, and then take the square root of that. So what you should get is that this is equal to 71.6 meters across the river. Alright. That's it for this one, folks. Let me know if you have any