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# In geometry, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. The three main trigonometric functions are the sine (sin), cosine (cos), and tangent (tan). The ratios of the sides are functions of the angles, and they are known as the sine ratio, cosine ratio, and tangent ratio. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

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### Video Transcript

Hey, guys, Mr. Clambered here. We're gonna talk about trigonometry today, and that's a word. A lot of people, when they hear they grown, they get, you know, uh, intimidated or scared by. But we're here to this breakdown. What is trigonometry and hopefully make it a simple as possible. And really, what it comes down to, trigonometry is the mathematics of triangles. That's one way I like to say it. Um, And then when we're talking about Trig and Trig ratios specifically, I just want you to think of the word ratio. Trigonometry really is about ratios you've heard about. So Quito A probably in these words like sine cosine tangent. And if you happen, that's okay. But trigonometry when it when it really comes down to it is essentially the mathematics surrounding triangles. And I won't talk about the basics of trigonometry. And that would be your three basic trig ratios that would be signed S i n e. There would be co sign. Pardon me and the other. The third would be tangent. Okay, now a lot of us, when we hear the word trigonometry, we think about right triangles. And there is a difference between trigonometry and right triangle trigonometry, right? Triangle trigonometry is a very specific type of trigonometry, and that's actually the trade that we're gonna focus on right now. And another course, you would talk about how to apply what used to be or what you would think. Could Onley be applied to right triangles? And you would apply that to any triangle? Right now we're gonna focus specifically on right triangle trig. And so the ratios that I just talked about Sign Coast unintelligent or these words that I've just mentioned are going to be specific to a right triangle. So first of all, let's draw a right triangle. Let's label it a be See Now if I'm gonna, let's say reference an angle, I could call like that angle X. I could also called angle A. I could also call it angle like feta. It's like this is a Greek letter later. Okay, so you might see I'm calling it X. You might see some textbook called Theta That's spelled T h e T A. You might also called just angle B A C, or they might just called angle A. But either way, whatever angle we're talking about in this case, I will mentioned angle theta by picking a certain angle to talk about the other three sides, then have a special name in reference to the angle. The high partners is always the high partners, and we know the hypotenuse is the side directly across from the right angle. The side opposite of theta, meaning it's not part of the angle theta is constructed by is what we call the opposite side and the site that's next to angle a or one of the rays that makes up angle. A. That's not the iPod news We call this side of the adjacent side. So again, depending on what angle you're referring to, you might have different sides label differently. So, for example, if we were referring to this angle up here and will be, the high partners would still remain the hypotenuse. But the opposite and adjacent sides would flip flop because a C would now be the opposite side. In reference tangle be inside. B C would now be the side that's adjacent to or next, or helps comprise or create angle B. So it's all depending upon where you located, so it is literally all relative okay, So we talk about the trig basics and you have these words sine cosine and tangent Trig and right triangle Trig. I just want you to think of it as ratios, and this is not something that was discovered. Like pie pie is a universal constant of taking the circumference of a circle and dividing it by the diameter. And you get this 3.14159 number and so forth it never repeats, never ends. That's a discovered, naturally occurring thing. The definitions of San Cosa contingent aren't a discovered. There were a defined ratio at some point Back, back, back in the day, mathematicians said we're going to find these ratios to be named sine cosine and tangent so forth. There's lots of them, by the way. But we focus on these three, and because it was so, triangles are so prevalent in nature and and build building and in art everywhere. So certain ratios popped up very frequently so frequently that we give them names. So the sign of an angle. Okay, Now it's abbreviated, by the way, s i n. And this is pronounced the sign of theta, not sin. Okay. S I n is pronounced sign and this is a sign of theta. The sign of data is defined to be or the sign of an angle is the ratio of the opposite side to the high partners. So literally you're just taking the length of the opposite side and dividing it by the length of my partners. So we talked previously in a previous video. I had a right triangle and I talked about Pythagorean triples, and I said that 345 was a very common thank you re in trouble. So if we talk about this 345 right triangle, we have theater right here. Then, in this case, the sign of theta would be four fifth. So it would be literally the opposite side length divided by the hypotenuse. That's how you define the ratio we call sign the ratio we call cosine. So the coastline of okay data, which we're going to abbreviate as just c. E. O s. So it's called the co sign of data that's equal to the adjacent over the iPod news. So in our example, that I drew the coastline of theta would be three fifth because the adjacent side a theta is three in their partners, obviously, is five okay, That leaves us with the third definition. And that is Tangent Tangent. Okay. Abbreviated t A n. So that would be pronounced. The tangent of Fada is defined by the opposite side over the adjacent side. So in our particular example, the tangent of theta would be opposite, which is for divided by the adjacent, which is three. Those are the three definitions. Meaning we do not go out nature, find them and say, Look at this, Let's make sense of it. We just went ahead and said Thies ratios are so common and lots of things we do with mathematics and applied mathematics. We're gonna give them names. They all have. They have Latin and Greek, you know, route roots of why they're called that. Um but we have signed S I n e co sign C o s i n e in tangent ta mgmt, abbreviated s I n C o s and t a n respectively, which are just definitions of ratios depending upon a right triangle and the angle you're referencing. So if we look at, let's say, just do one more example here at a page here, right triangle. And let's say this is five and this is 12 and this is 13, which is another common without your in triple 5, 12 13. And I'm gonna look at this angle right here, which we're gonna call fee. So I'm just trying to get you used to all kinds of different symbols. This symbol right here is P H I. The Greek let her feet. So if I said Hey, what's the sign of feet? Remember, Sign is opposite. Opposite This is the opposite side of the triangle over the high pot news. There will be 5/13 if I say what's the co sign of fee? The co sign is the adjacent over the high partners. That would be 12 over 13. And if I said, What is the tangent of fee? The tangent is the opposite over the adjacent by definition, and that would be 5/12. Those are your three basic building blocks off all trigonometry. Not just right triangle trip, but we'll start with right triangle trick and then maybe we'll work our way up to none. Right? Triangle trick. But these are the three basic building blocks definitions that we have we just at some point agreed upon, you know, hundreds of years ago and we have the three basic ones sine cosine and tangent. I'll see you next time, and we'll talk about how we can use trigonometry to help us. Thanks. Webster University

#### Topics

Rigid Motions (Isometries)

Volume

Terminology

Relationships Within Triangles

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