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RC
Numerade Educator

# Trigonometry is the branch of mathematics concerning the relationships between the sides and the 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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### Video Transcript

welcome to our math review video on trigonometry. In this video, we'll be talking a lot about right triangles, right. Triangles show up all over the place in physics, which is why we're gonna have a special review video and, in fact, several examples on how to use them. Um, we're also gonna talk about Trig and metric functions in this video. Reintroduce you to those because they can also show up, especially when we're talking about any sort of periodic or harmonic motion. So think of a pendulum swinging back and forth anything like that. So let's go ahead and get started. Right. Triangles, if you remember, are triangles that have a 90 degree angle in them. So these triangles air particularly helpful because they can have two legs that are perpendicular to each other. And this will come up when we talk about vectors. So, for example, of if you remember from your whatever physics you may have taken before, a vector looks something like a line with an arrow at the end, it's supposed to depict some sort of magnitude that's pointed in a particular direction. So, for example, if we walk 5 m to the right and then say we were to right, turn and walk 5 m up Then here if I were to draw a vector, showing how we went from the beginning directly to the end. You see, I've actually created a right triangle and this interaction here is why right triangles become so important for us later on. So let's consider how all of this is related. So, um, let's give some names to our sides here. I'm actually going to just name one of the angles theta here because the other angle then will necessarily have to be 90 minus data. Remember that inside a triangle, all of the interior angles must add up Thio equal 180 degrees. Since we're requiring that one of those angles be 90 that means the other two must add up to be 90 degrees. So if I say that I know this angle, it means the other one has to be 90 minus. Whatever that angle happened to be. So looking at that, then the first relationship you probably remember is the Pythagorean theorem. A squared is equal to B squared plus C squared where a is something called the high pop news in BNC or what we call the legs. So this is a nice relationship for how the sides related, and we already looked at how the angles air related. What we're really interested, though, is. And how can we relate the sides to the angles and vice versa. So the way we're going to do that is with what are called trig and metric functions. So you probably remember some of them. For example, we had sign of data or sign, which we can write as sign of data. And we have cosine, which we can write as coasts Fada. And then we had tangent, which can be written as 10th data. These were are three principal triggered a metric functions, and in fact, 10th data is simply sine theta over cosign data. So sine and cosine data are really the heavy hitters when it comes to trigger the metric functions. Um, so looking at these, though, how do they relate the angles to the sides? Well, it turns out that the sign of data is equal to the ratio of the opposite side from the angle theta divided by the high pot news. So, looking at that, then the opposite side would be be so. In this case, sign of data is equal to be over a cosine theta, on the other hand, is equal to the adjacent side to the angle, divided by the high pot news, which in this case then is going to be equal to see divided by a. And looking at this, you can probably figure out with a little algebra that since 10th it is equal to sign over coastline. That means it's equal to the opposite side, divided by the adjacent side. Or, in this case, we have B oversee. So, um, here's our different relationships. Sometimes people use a pneumonic device to remember this. They call it so Cartola, which is to say sign is equal to the opposite over the hypotenuse. The coastline is equal to the adjacent over the high pot news, and the tangent is equal to the opposite over the adjacent. Um, it's not a bad pneumonic device. I definitely recommend that you remember it, and knowing these three setups, we can see that we have successfully related the sides to the interior angles. Thea other thing that we might want to remember that this is another. There's a couple other relationships here. One in particular that shows up is something that's known as the law of Cozzens. Okay? And what the law of co science does for us. It says if we have a generic triangle Um, sorry, we have a B C and then the opposing angles air labeled as capital a Capital B in capital. See? See what I've done here is that the angle opposite to the side has the capital version of that name. If we do that, then we find that a squared is equal to B squared plus C squared minus to B C. Cosine of a. Okay, so here's another relationship that can end up being used. Um, this shows up in rotational motion a little bit, though I may or may not explicitly use it. Some classes don't worry about it. Others do. But there it is for you to remember. Some of you may have seen it before. Okay, so let's talk a little bit about what sine and cosine look like when we plot them. Because these are actual functions. They're not just those ratios. They're not just static, their actual functions that we have So the sign of X, where X is our independent variable turns out to look like this. It oscillates back and forth. And in fact, if I were to extend this axis, it would continue to oscillate back and forth and it oscillates back and forth over here and again. If I were to continue to extend the axis, it would continue to follow that pattern forever and ever out to infinity. Um, so this sort of motion from a function is really helpful. And then cosine is extremely similar. Except instead of starting at the origin 00 it's going to start at X equals zero and y equals one, which is the amplitude of these functions. That is to say, sine and cosine are never larger than one, and they're never smaller than negative one. So negative one is there a minimum and one is their maximum. And this is really helpful because it means that if we were to multiply either of these by some constant, say a, then A would become the new amplitude of the function f of X in this case. Or I could do the same thing with Kassian. Notice also that really sine and CoSine are just displaced by some amount. If I were to take sign and shifted to the left just enough to where that peak was, that was on the Y axis, then it would look exactly the same as co sign, and that's by That's intentional. That's that's a very helpful property as well. Um, tangent because it's equal to sign over CoSine. We'll also have a repetitive motion. But notice that since it's over cosine cosine periodically hit zero. What's gonna happen is that tangent is gonna have a whole bunch of vertical Assam totes that it can't cross. So it looks something like this motion. Here we'll go more and more into what these different plots mean later on, Um, I should note that there are inverse functions toe all of these. So Sign has an inverse function, which is the Ark sign of X, which can also be written as signed to the negative one of X. The reason I don't like doing that is because sometimes it gets confused with one over sign of X, which is actually called the co Seacon. And that is not what we're looking for. A sign of X I will use it as being equal to sign to the negative one of X. Similarly, we have co sign of X, which has an inverse function that's a coast of X or co signed to the negative one of X and tangent has a similar inverse function. Tan of X is a 10 of X or 10 to the negative one of X. And all these really mean is that if I take sign of X and then I operate on that with the inverse sign, then I'm going to get X out of that. They're inverse operations. Now, there's one more thing that we should take a quick look at just to be prepared for when we come across inclined planes. And that is, if we have two angles that are arranged like this, where the rays of these angles are perpendicular to each other, you'll find that these two angles must be the same. This comes about because of the perpendicular charity here. Okay, because this is 90. That means that this is 90 minus data. So if this is data 90 minus data, this has to be 90 minus data. Those with same angle, which means this is data. This would be really helpful when we get to inclined planes

RC
University of North Carolina at Chapel Hill
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Simon Fraser University 