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Trig Word Problems and Applications

In mathematics, trigonometry, also called triangulation, is a branch of mathematics concerning the relationships between the sides and the angles of triangles. Trigonometry is used in the measurement and description of the shapes of objects, such as the positions of stars and the sizes of galaxies, as well as in navigation, engineering, and physics. Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles. The word "trigonometry" comes from the Greek words "trig?non" (????????, "triangle") and "metron" (??????, "measure").

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Video Transcript

Hey, guys. Mr. Kleinberg here today, we're gonna work on some trigonometry word problems. So we're going to see how we can use trigonometry to solve some really world problems. Um, what's often very helpful is drawing a picture in my picture. I mean, distraught triangle label what you know, and then you're gonna find out it's a lot like the other problems we saw, um, before in the previous videos. So let's do a couple. One says we got a boy flying a kite. Okay, Lets out a string that's 300 ft long, and it makes an angle of 38 degrees with the ground. So we're gonna assume or negate the child's height because, technically, the child would be off the ground a little bit. But we're gonna a lot of times in these problems we negate that. Um because we're looking for is if this is where the string is released, I'll do that in different color. If this is where the string is released, then that distance is 300 in this angle is 38 degrees. So then it says, assuming the string of straight how high above the ground is the kite Well, again. Technically, we'd have to know how tall the kid is. So let's say this is where the kid is. Once we find the height, we technically have toe. Add on that distance to figure out the true height of the kite. So if the kid was, let's say, 3 ft or 4 ft or whatever, we had adding that to the answer. So some problems are a little more picky, Um, or a little more specific. This one doesn't mention it. So we're gonna assume that we were ignoring the height of the child, or it's like the kites being released from the actual ground. So sometimes you have to kind of get rid of the what ifs. But we have this triangle now notice we have a right triangle situation, and based upon our angle, we have an opposite. We have a high pot news and remember that opposite and hypotenuse are related to sign, so I could say All right, the sign of 38 degrees equals the opposite, which I'm calling each year. That's the height of the kite over 300. And if I multiply both sides by 300 these will cancel. And on my calculator. I'm making sure I'm in degrees. I'm gonna take 300 I'm gonna multiply it by the sign of 38. And you're going to get 184 0.7 ft is roughly the height of the kite. Okay, let's do one more or a couple more, actually, number two. Ah, letter leaning against the wall makes an angle of 74 degrees. With the ground here, it makes sense of This is the ground. This is the wall. This is the letter because the ground should touch the floor. It says 74 degrees with the ground and says if the foot of the ladder is 6.5 ft from the wall So now we know that this distance down here is 6.5 ft. How high on the wall is the letter? So now we're looking for that value right there. And I could have called H for height or anything. I decided to call it h or so again, you're gonna look at your angle. You're gonna go all right? I'm looking for the opposite. I have the adjacent That's got tangent written all over to the tangent of 74 degrees is equal to your opposite side. Over. You are adjacent side. So if I multiply by 6.5 again on both sides, those were canceled. I go to my calculator 6.5 times the tangent of 74 I get 22 point seven ish feet is roughly how high up the wall we are. Okay, so there's lots of different questions we could ask where you're essentially going to be even an angle aside, and then you're gonna try to find a missing side, and all you're gonna do is relate those three and determine Is it a sign situation, coastline situation or attention situation? There are some other questions could be asked, and that would be something like, Look at the question. Nine. A latter leans up against a building. So this is a very common question because it's easy to make a triangle with a ladder leaning against the building. So we have a letter leading up against the building. Here's my right triangle. The top of the ladder reaches a point on the building. It's 18 ft above the ground. You know this is 18, says the foot of the letter is 7 ft from the building. So there's seven it asked for. Find the measure of the angle, which the letter makes with the ground. So it's asking for Hey, what is this angle gonna call Data now? I could have asked a lot of things. I could have said. How long is the latter? What we could do a little Pythagorean theorem, right to figure out the latter. I'm gonna say that's l squared. So if we were interested in that, that's not the questions asking. We could use a figuring out there and to do so. Um, but in this case, we're looking for the angle looking for this angle right here, and we have the opposite and we have the adjacent So that's like saying all right, tangent of theta equals the opposite over the adjacent. Remember, this was trying to solve for an angle. We have to actually do an inverse toe, undo the tangent. So we're gonna do inverse tangent equals inverse tangent. Tangent equals inverse tangent of 18 7th These cancel your left with data and I'm gonna go to my calculator and I'm gonna do inverse tangent 18 divided by verse, tangent part me 18 divided by seven and it's gonna give me roughly 68 7 degrees. Again, I'm setting up setting up a right triangle. I'm labeling the two out of the three things that I know, and then I'm just going to set up a trick equation. So let's get eight. In order to reach the top of a hill, which is 250 ft high, one must travel 2000 ft straight up the road. So the hardest part sometimes is recognizing what's what. The hill has an elevation of 250 but you gotta travel 2000 ft to get there, it says. Find the number of degrees contained in the angle, which the road makes with the horizontal. So the horizontal is this part of the triangle. So again, they're asking for what is the angle made with the ground. So I'm gonna call it fate again. Here's my opposite side. Here's my high pot news that's got Sign written all over it. So I will be the sign of theta equals opposite, which is 2 50 over High Partners, which is 2000. Once again, we're gonna have to undo that sign with a oops in verse sign. I'll inverse signed the right side as well. Again. These were going to cancel or left with faded equals and I'll go to my calculator. I'll inverse sign 2 50 divided by 2000 and we're left with a measurement of about 7.2 degrees. So not too bad. So there you have it. There is how you can use trick to solve some rubble. The problems. Now let me show you. Actually a problem that z a little more complicated than this, but essentially, you know, uhh same. Set up a couple more steps. So let's say I have the situation where I have a building over here. Okay, this is this'll is a building. And let's say that building is 700 ft tall and I have two people looking at it. I have person one looking at it from here and person to looking at it from here, and I know um, the angles of which their eyes air making with the horizontal to the top of the building. So let's say I know this is, let's say, 38 degrees on Let's say I know this is 52 degrees My question is how far apart it's observer one from observer to well, a couple things to note. This triangle here is not a right triangle. We have not learned about Trig other than right triangle trick. So there's nothing I could do with that triangle that would relate right now with Sankoh Scient aged Nothing. Those rules right now on Lee, apply two right triangles. I do have this right triangle if you notice. And I also have the bigger right triangle like this big right triangle here. So if I knew hear me out. If I knew this distance and I knew this distance the night could subtract him to find the distance that I that I want. Well, okay. Looking at that green triangle First, let's call this distance here X, which is this distance here. So this distance right here, we're gonna call X. Well, in regards to 52 I have the opposite. Here's the adjacent. So would you agree that hopefully tangent 52 equals the opposite which is 700 over x. Multiply both sides my ex The excess cancel we're left with X Tangent of 52 equals 700. I'll divide 700 divided by Tangent of 52. And in my calculator I'll do 700 divided by tangent 52 When I find out is that X is approximately I'm just gonna go to the nearest whole 500 and 47 feet. Okay, Now what I'm gonna do is the same thing. But I'm gonna do it for the entire distance. Uh, his entire distance. So this distance here, I'm gonna call that. Why? Well, in that triangle, here's why. Here's 38 and here's that 700. So just like we did here, I could say All right, The tangent of 38 equals the opposite over why and in solving it in similar fashion, where multiply both sides by y, then divide by tangent of 38 we get 700 divided by the tangent of 38 degrees and with why is roughly again to the nearest whole 896 feet. So now if we want to find this part that's missing a k a. The distance between them. I could take 8 96 and subtract 5, 47 8, 96 minus 5. 47. And the two observers are roughly going to be 349 ft apart from each other. This is essentially to trick questions in one. So hopefully you enjoy this. I'll see you next time. Have a great day.

Webster University
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