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Using Trig to Solve Triangles

Trigonometry (from Greek trigonon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is also the foundation of surveying.

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Cairn University

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University of Michigan - Ann Arbor

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University of Nottingham

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Okay, so we're here today to talk about how to use the trigger. Metric ratios Sign coastline contingent to find a missing side, find a missing angle or do what we call solving a triangle. And solving a triangle is not just finding a specific angle or specific side. It's finding all of the missing sides and angles, so let's start off with this example. Nine. An example. Nine. You have a mark side BC, which is 11. You have the Angle B, which is 37 degrees an angle, or the side that we're looking for is site X, which is the hypotenuse. Now. What you need to do on any one of these problems is a side based upon the angle. This is the important part based upon the angle. What side and or angle do you have, and what triggered ratio would they get you? So the angle we know is this 37 degree angle right here we have the adjacent side, which is 11. Pardon me, we're looking for the hype oddness. So what? Trig ratio relates the adjacent in their parties and that would be co sign the cosine of 37 degrees the cosine of the angle equals your adjacent side, which is in this case is 11 over X. Now we have an equation, and now we can solve. So what I'll do is on multiply both sides backs. Now I have X Times CO sign of 37 equals 11. Since the X is canceled out and they'll divide by Cassandra 37. Keep in mind this. That's not cozying times 37. That's cosine of 37 which represents a particular ratio. So I'm gonna divide by that it's gonna be 11 divided by co sign a 37. So I'm gonna use it. Calculated this point making sure my calculator isn't degree mode and I'm gonna do 11 divided by the co sign of 37 degrees. And this is going to give me X is approximately 13.8 when a round to the nearest 10th. So again, here's what I'm gonna dio like my triangle and I'm gonna ask myself, what angle do I have and what sides do I have? And I'm looking for in relation to that angle, and it's always going to set up the problem. It's gonna create an equation we're gonna solve for it. So look, a question 10 here. So, in relation to this angle, we have the adjacent side. We're looking for the opposite side. Well, that would be tangent. So tangent is remember, opposite over adjacent. So the tangent of 32 in this case is equal to the opposite side, which, we don't know over. The adjacent site was just 13. In this case, I'll multiply both sides by 13. And so I have 13 tangent of 32 equals X again, going to my calculator 13 times the tangent of 32. Keeping in mind that I am degrees, I will get approximately 8.1 units his ex. So again, that's similar question you before. Is that where's your angle? What sides do you have? What you looking for? And you can set up your question from there. So if you look like a question 12 notice, I have this angle. I have the high pot noose, okay? And I'm looking for the adjacent that would be causing again so we could say that co sign of 60 equals the adjacent over the high pot news multiplying by 11 on both sides. These were gonna cancel you're gonna get 11 times the coastline of 60 and you're gonna get exactly 5.5 equals X Now, in a future video, maybe you've already seen it. By now, we're gonna talk about a very special type of right triangle, and that is called a 60 or 30 60 90 right triangle. Of course, this is 30 because that's what's left of 1 80. And there's a special relationship between the high pot news and the smaller side. And that is the hypotenuse is double the smaller side and notice that well, 11 is double 5.5. So it's it's something that's interesting because you could do this with the Trig and you can also do it with what we know about special right triangles. More on that later, which, by the way, is based upon the Pythagorean Theorem. So all this stuff in some way kind of tied together. Okay, we can also use Trig to work backwards and defined the angle. Instead of giving an angle on the side and finding a missing side, I could give you two sides and against you to find the angle. This is gonna require inverse operation, just like we would undo plus eight with this attraction of eight, and we would undo it times by 10. By dividing by 10 we have to undo a sign. A co signer a tangent By using this thing we call inverse sine inverse coast sign or inverse tangent. And again for now where you can use our calculators built in features. Do that. Um, so let's follow along here on question one. We wanna know this angle is now in reference to this angle. It looks like we have the pot news and it looks like we have the adjacent that would be coincided. So that co sign of that angle theta. Remember, I said data might be used. The casino theta is equal to the adjacent over the hypotenuse. Now, what's gonna undo cosine is this inverse operation which we call inverse cosine. And I'm gonna write it like this with this little one here that's not cosign raised to the negative won. This is referred to as you pronounce it as inverse cosine in verse co sign and by definition, it's gonna undo the casino, and what's left is a theta. But if I inverse cosine the left side that I need to inverse cosine the right side. So what I'm gonna do is squeeze in here inverse cosine of 12. 13. And again, I'm gonna go to my calculator where there is an inverse cosine button and I'll do the inverse co sign of 12 13. So essentially theta equals the inverse cosine of 12 13th, and that gets me a value of approximately 22.6 degrees. So we've done is we've set up the trick ratio again, like we know the coastline of what angle equals what side to what side except of the two. Sorry. Of the three things that we have to will be knowns. One will be unknown, and we're just going to solve for the missing one. Okay, so let's try another one. Question two. So in this case, you have fatal, which is this angle. It looks like we have the adjacent here in the opposite here. That would be tangent. So we should be able to say the tangent of data, which is unknown, equals the opposite or the adjacent. And again, we're gonna do is we're going to do inverse tangent, both sides. What happens is the inverse tangent in the tangent, canceled by definition. And then I'm gonna take this night calculator, and the inverse tangent of four or 13 is approximately. And I'm gonna change this symbol here to approximately because it's not exact 17.1 degrees. So this would be an example again, how you would use thean verse trick operations in order to solve foreign angle. So whether you're solving for a side or an angle, it's the same you're going to set up a trig ratio based upon the definitions is either sign is opposite over adjacent coastline. Part of the sign is opposite of our iPod news co sign is adjacent of our iPod. News for a tangent is opposite over adjacent, and you're gonna right that equation out and you're gonna have a unknown on one of those three positions. And then you're gonna use algebra to solve situation where it says something like Solve. Pardon me, it says, Solve the triangle. Now we're gonna find everything that's missing. Well, if you're given an angle to start with, it's easy to find the missing angle because that's this is what's left of 1 80 or out of 90 actually. But 1 80 minus 62 minus 90. Because I'm getting rid of 62 I'm getting rid of 90. And that's gonna get us an angle of 28 degrees right here. Okay. So again, I just did basically 1 80 minus 62 then minus 90. And they got us with 28. That's how you find the missing angle. Now we have this side. We need to figure out the side BC, and we got to figure out the side B A. Let's start with side BC. I'm gonna call it may get rid of this. I'm gonna call the site X. I'm gonna call this site. Why? Well, in relation to and you pick whatever angle you want now, which is nice. But in relation to 62 side B. C is the adjacent side. 22.6 is the opposite side. So that would be tangent because we have an opposite. And adjacent, So tangent of 62 equals 22.6 over X. Okay, remember, I can multiply both sides now, by X, we get X tangent of 62 equals 22.6 because the extra they're gonna cancel out here and then I'll divide by tangent of 62. And so I'll do 22.6, divided by tangent of 62. That's gonna give me 12, approximately 12 point zero, but to be 12.16 yada yada yada. I know it says nearest 10th, but to get a little bit more exactly now, I could have used the angle of 28 and then said, Hey, this is the opposite. This is the adjacent And I could have said the tangent of 28 equals X over 22.6. And if I multiply both sides by 22.6, believe it or not, 22.6 times the tangent of 28 is going to get you right back to that value. So the cool thing is, it doesn't matter what angle you reference it in it. Just whatever angle you pick is going to determine a different, um, possible ratio, um, or equation, because the sides we're gonna relate to each other differently, depending on the definition. Now notice. We're still looking for why we could do another trig equation. For example, we could say All right, I have angle 62. I have the opposite. Here's the iPod news. I could do a sign equation and solve Fort Or I could do with that jury in theory, because I have a right triangle and I have two of the sides because we just found X. So technically we could use what exes and what 22.6 is and do the Pythagorean theorem to figure out the missing side. Or you could just do another trick equation. So, for example, I could say right, if I'm gonna trick, I could say something like the sign of 62 equals 22.6 over. Why? Then I could solve for why? Or I could say X, which is approximately 12.16 squared plus 22.6 apartment six squared equals. Why squared solve both those those equations, you will get the same answer. Okay, so at this point, after you've figured out two of the sides and you have all the angles, then you can use trigger again to figure out the missing side, or you can use Pythagorean theorem. So in any situation where you ever boiled down to a math problem, that you could solve in multiple ways. You don't have to solve it using the trick because we're talking about trigger right now. You could solve it any which way you like a ZA, long as you're using legitimate, legitimate mathematics. So let's go ahead. Do one more example over here. All right, so and question 18, I'm always gonna start with finding out the missing angle. It's always the easiest thing. So I'm gonna do 1 80 minus 90. Pardon me. Minus 90 minus 51. And I'm gonna get 39. So this is a 39 degree angle. Okay. Okay. So we know what we know. Our two ng we know all the thing was, actually, So now I'm gonna label this side X, and I'm gonna label this side. Why? Okay, so if I'm going to solve for site X and I'm gonna use, let's say angle 51 this is the adjacent This is the opposite. So the tangent of 51 equals the opposite over the adjacent. Okay. Now, I could have also used this angle. So X is the opposite. Nine is the adjacent. So I could have said the tangent of 39 equals X overnight solving both these equations, you will get the same thing. In fact, I'm gonna use this version because now all I have to dio is multiplied by nine on both sides instead of having to rearrange the equation. And nine times changing of 39 equals approximately 7.3 his ex. Now that I know this is 7.3, I could say nine squared plus 7.3 squared equals my partners. And that would be 81 plus my answer or 7.3 squared and then square rooting that answer. And we get roughly 11.6. Okay, One more example here were given on 20 the high partners. So once again, I'm gonna find my missing angle rather quickly. This is 37 degrees. That's just the difference from one getting rid of the 90 degree angle in the 53 degree angle. Now in reference to 53. But call this X and Y and I want to solve for X. First five is the hypotenuse X is the opposite that would be signed. The sign of 53 equals the opposite over five, multiplying both sides by five. I get five times the sign of 53 again, these you're gonna cancel. And so you're gonna get 3.993 ish or in other words, roughly four equals X. At that point, I could use the Pythagorean theorem to figure out why, or I could do trick again. And I know it's gonna be roughly three. Why is gonna be roughly three? Because if you recall, a 345 is a pathetic Dorian triple where this is five. We got something be really close to four. So this has to be really, really close to three. So here's an example of how you could use trig, um, to solve the triangle, which means to find all the missing pieces. Or if you just wanna find one particular side of one particular angle, you could do so Remember, if you ever, ever, ever, ever, ever going to find an angle, you're gonna have to use inverse trick. Okay, I'll talk to you later.

Webster University
Top Geometry Educators
Christine G.

Cairn University

Heather Z.

Oregon State University

Kristen K.

University of Michigan - Ann Arbor

Samuel H.

University of Nottingham

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