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Trigonometric and Exponential Functions - Overview

In mathematics, trigonometry 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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Okay, so this is a picture of the unit circle, and I want you to take a deep breath, don't have a panic attack. I just show this to you because we're going to start talking about tricking the metric functions. And even though the unit circle is a scary concept, it's actually the best way to think about the trig and metric functions. And it's also the best way to define the trig and metric functions. So what we have here is a circle of radius from one centered at the origin and now thes angles. If you're more comfortable in degrees, that's fine, but I'll probably stick with radiance. Thes angles are from the positive X axis. So here's the positive X axis, and so my angles, we're going to be increasing counterclockwise. That's important from the positive X axis. So I'm going to hit 30 45 60 91 20 thes air, Just some special values. But my angle it's very important, is just how much I've rotated from the positive X axis counterclockwise. Now, if I take a ray at one of these angles, it's going to intersect the unit circle at a point now the X coordinate of that point is to find to be the cosine of the angle, and the Y coordinate is defined to be this site that ankle. Now you can also find sine and cosine using right triangles, and this really is the same thing. But it's slightly more general because we're going to actually allow our angle to increase or decrease indefinitely. So we want to notice just a few things about the trig and metric functions. First of all, there are other trig and metric functions, but they're all defined in terms of sine and cosine. So recalled that we have Tangent of Fada, but that's just equal to signed data divided by cosign data. So there's really not a lot of new information about tangent. Same thing with seek it. Seeking is just to find to be one over CoSine Data Co. Seacon is defined to be won over signed data and contingent data is just to find to be cosign data over signed data. So these were just other triggered a metric ratios that sometimes they're easier to work with, but notice they really all depend on signing coastline. We'll also note that attention is just one divided by tangent data. Soco Tangent is sometimes called reciprocal. Sign seeking is called reciprocal co sign because it's just the reciprocal go seeking is reciprocal sign. But even more, it's true. Not only are all the other four trig functions to find in terms of sine and cosine, sine and cosine really aren't that different. And what I mean by that is that if we look at the unit circle, we can notice that if I take an angle, let's say 30 degrees and I look at the X coordinate of the intersection. So I look at the co sign of 30 degrees square to three over to, and I add 90 degrees. So I go upto 1 20 then the sign of 90 plus that original angle is going to be equal to the cosine of the original ankle. So what I mean is that sine theta is really nothing more then just cosign of theta plus, however, to or if you prefer 90 degrees so sign is really just a translation of Kassian, which is really cool because really, if we could just understand the basic properties of one of the trick functions, the other one's really you're going to follow. And that's something we're going to see in Calculus is, Well, once we figure out how to do limits with Sign, take derivatives with sign, then really the rest is gonna follow pretty easily. So this is just an introduction again just to remind you of the basic trig functions sine cosine and then the corresponding other trick functions there, defined in terms of sine and cosine, this sort of key identity relating signing co sign. And all of this is just coming from symmetry in the unit circle. So as we move forward with the trig and metric functions, it's gonna be very important for you tohave either this unit circle in the back of your mind if you're really comfortable with it, or maybe even find a copy online or in a book, see that you can reference it. So I told you that most of the trick functions actually, all of the trick functions are defined in terms of sine and cosine, and that we can think about CoSine is just a translation of sign. So I'm going to talk about just the properties of signed data just for a little bit just to give an introduction. So, first of all, the domain of signed data. So if I define f of data to be signed data the domain of dysfunction Well, recall that an angle on the unit circle can just extend indefinitely if I go counterclockwise and it can actually extend negatively if I go clockwise. So the domain for my angles really is all real numbers. But now, remember that science data was the Y coordinate of that intersection point on the unit circle. So the largest Y coordinate on the unit circle will be one, and the smallest will be negative one, and we're gonna hit everything in between. So the range of sign is going to be negative. One toe one. This is going to be a very important fact. That sign will live in this sort of small interval between negative one and one. The same is true for co sign. So let's think about sine theta and whether or not it has an inverse whether or not it's 1 to 1, we'll notice right away that sine theta is not one toe one. And again, if you think about the picture of the unit circle. It should be clear that sign of data is gonna be equal to sign of data plus two pi. Because if I add two pi to an angle, I end up in exactly the same point in which I started, which will have the same y coordinate of that intersection point. So sign is not 1 to 1. So we need to restrict the domain to make sign data 1 to 1. Now, there are more than one ways to do this, but there's a standard way to do it. And so what I'm gonna do is I'm going to restrict the domain to negative prior to two pi over two. And if you're asking me why, Well, someone smarter than me said this was a way to do it, and it's really just conventional. A lot of things we'll see from algebra or just conventional. We just want to think about it that way because then we have a kind of consistent way Thio work on problems. Okay, so on this domain sign is 1 to 1. And if you want to see why, I just look back at the unit circle as the angle goes from negative prior to two pi over two. We're getting a unique Y coordinate as we move from those two angles. So on this domain we can define FM verse as sign members of favor. And sometimes it's also called arc sine. Either way, it's perfectly fine. Arc sine is sort of to remove the ambiguity about this negative one that it's not actually a reciprocal the reciprocal of scientists Cosecha. This is actually the inverse of Sign the function in verse. So when I compose sign and sign in verse cancer naturally and so I'm not going to talk about the rest of the trick functions. But we can do a similar domain restrictions, and we can to find the inverse of those functions in a similar way. But we're gonna have an entire section later on when we talk about derivatives of in restrict functions and we'll just kind of highlight all of the things we need to know about the other trick functions at that time. So if you remember from pre calculus, one of everybody's favorite topics is trig identities, and I have a hard time remembering the trick identities that there's a bunch of them, ah, lot of them are more or less the same. Some of them are different. I've listed three here that I think kind of encompass everything you need to know. And I'll tell you that if there's one trick identity I would memorize and not forget, it's this one. And it's actually not too hard to memorize because it really is just the Pythagorean theorem. If you look back on the unit circle, the X coordinate and the Y coordinate form the legs of a right triangle with hypotenuse one because you're on the unit circle. But these other two are also important as well, and their thesis? Some and well, there's some identities or angle addition identities for sine and cosine. That air also really important, and we'll see how these can be applied and just just a minute when we do some examples. But also, first of all, I want to notice something else from the unit circle, and it's also going to give us a chance to introduce another idea that we need to introduce about functions, and that's the concept of even and odd functions. So notice from the unit circle that if I look at the negative angle. And so what I mean by the negative angle is if I look at an angle going counterclockwise, look at the corresponding angle going clockwise. Notice that the cosine of those angles will be the same. So if I make the argument of cosign or the angle negative, that's not going to change the X coordinate. It's just gonna be a reflection over the X axis. The X coordinate doesn't change, but the Y coordinate will, because I'm reflecting over the X axis when I make the argument negative. And again, just look back at the unit circle and you'll see what I mean. If I take a counterclockwise angle and the same clockwise angle, uh, from the positive X axis, the Y coordinates are going to be negative. One is going to be positive, and one's gonna be negative. So a function f is called, even if it has this property that Kassian is illustrating. If I make the argument negative, it doesn't change the value of the function, and then a function is called odd. If it satisfies this property, that sign satisfies if when I make the argument negative, it makes the entire function negative and these ideas, they're going to be important there. You could probably live without them, but they make a lot of problems later on Really easy if you can remember certain properties about even in odd functions. And so cosine and sine are kind of marquee examples of even and odd functions. Other examples of even functions are power functions with even exponents. And then, of course, that maybe that's where the terms even if not come from. So for instance, X squared is an even function, whereas X cubed is an odd function. And then any power function with an odd X moment is gonna be odd. So there's another couple examples here, so this is a lot of information. But, you know, all of these things are gonna be useful later on. And even if we don't memorize them now, we want to be able to reference back some of these trigger metric identities and some of these basic properties of the trig functions. So this is a table of some common trig and metric about use. Now, where did these come from? You may ask. Okay, Well, where does pirates six piper for Piper 33 pirate, 475 or six. What's so special about these angles? Well, they're actually coming back from 45 45 90 triangles and 30 60 90 triangles. So if you remember, if you have a triangle a right triangle where these two sides air concurrent. So these two angles or congruent right here This is called a 45 45 90 triangle. And so this angle 45 is really pirate before so 45 degrees or pirate before and we have some special values. So if this leg is A, this leg is also going to be a in this leg is going to be square root of two way by the Pythagorean theorem. So we had this sort of special relationship with 45 45 90 triangles and that's where these trig values were really coming from. So pi over four, this one over route to is really coming from this route to in the 45 45 90 triangle. Similarly, we had a 30 60 90 triangle. So that was where this angle it was 30 degrees. And in that case, if this was say a okay, then this was going to be twice a and then this was going to be square root of three A. And then that's where these values air coming from here for pi over three. Okay, so that's 30 degrees pi over six is really 60 degrees, so this angle is gonna be 60 degrees. So these special values were just coming from 45 45 90 triangles in 30 60 90 triangles and using that Pythagorean identity as well. So you know, if you're wondering, where does this to a come from? Where does this square to? Three a come from? It's coming from the Pythagorean identity that Okay, if I square this and I square this, it better be the square. Same thing here. If I square this and I square this Adam together, it better be. That's great. So again, it's useful toe have a table. These values, whether or not you need to memorize them, is kind of up to you. And maybe whether or not that's required for you, I'd say that it's worth memorizing because thes values come up a lot. But if nothing else, you should have a table of them handing when you're solving problems. So the last thing we want to review from pre calculus, our exponential functions. So we've talked about polynomial is We've talked about tricking the metric functions just briefly anyways, so let's talk about exponential functions in exponential function. Well, first of all, it represents exponential growth, hence the word exponential. But an exponential function has the following form. It's a fixed number, a raise to a variable X power. So here a is going to be a real number, but we're not going to just let it be any real number. It needs to be a positive number other than one. So a is greater than zero and a is not equal toe one. And the reason we require that a not be equal toe one is not necessarily clear right now. So if I let a be one well, one to anything is just gonna be one. So it seems like we would just get the constant function one. But we want to exclude that case because we actually want exponential functions to be 1 to 1 or in vertical. And so just a couple of notes about exponential functions. The domain of an exponential function is all real numbers. There's no problem in the domain in the range of any exponential function is going to be 02 infinity, not including zero. So just the positive numbers. So another important thing to note about exponential functions is there in behavior. We're going to break this down into two cases. So the first case is when a is strictly greater than one, and in this case, our function is really growing exponentially. So it's starting at zero down in negative infinity, and it's growing, growing, growing, growing. And when we get to the positive numbers, it shoots off to infinity really quickly. It grows exponentially, so it's X goes to minus infinity. The function is going to zero and his X goes to infinity. The function is going off to infinity very rapidly. Another thing to note is that exponential functions cross the Y axis at one. Because if I plug in zero for X, anything to the zero power, it's just morning. Now. If a is less than one, the exact opposite thing happens. The function is actually going to exhibit exponential decay, and so it's actually going to start very large when it's negative and it's gonna decay off. 20 His X goes to infinity. So now is X goes to minus infinity. The function blows up to infinity very fast, but his X goes to infinity. The function actually levels off very slowly towards zero, never quite reaching zero. So you can see that these two cases air just reflections of each other. Okay, so here we have some basic properties of exponents and these air really important when you're dealing with exponential functions to remember. So for instance, if I take the this number A So the number A is sometimes called the base the base of the exponential function. So if I take the base and I raise it to the sum of two numbers, that's the same thing. Is just taking the base thio each number and multiplying them together. And if I take it the base to a number and then raise it to another number, that's the same. Is raising the base to the product of those two numbers. If I take to different bases A and B and multiply them together and raise that product to a number that's the same is just raising each of the base to that number and multiply them together. Then finally, if I make whatever the exponents is negative, that's the same. Is just taking the reciprocal. So 1/8 of the X. So these were just some important properties to remember again, you probably have to review them, which is fine again. The most difficult part is just remembering this massive amount of information because you'll be doing a problem and I'll say something like, Oh, yeah, well, you just do, um, you know, if I have to to the se X plus Why and I say that's to the X Times to the why you may think, Oh, where does that come from? Well, I'm just using these properties that are probably somewhere deep in the back of your mind, so just remember that they exist and will reference them when we need them. So let's talk a little bit about the fact that exponential functions air Oneto one, and this is a result that you can actually show just using the properties of exponential functions. But it's very useful because remember the reason we care that a function is one. The one is that means that they have in verses and so this is actually going to bring up logarithms, and so maybe you haven't made this connection, but it's a very important connection that logarithms if you can understand exponential functions, logarithms air simply just the inverse of exponential functions. And that's really the best way to think about them. So recall if I had my function A to the X with domain, all real numbers in the range. 02 Infinity, then, because it's one the one, there's a numbers in the inverse. Now, remember this number A is called the Base of the Exponential. It's also going to be the base of the logger rhythm that is the inverse of that exponential function. So this is Log Base A of X, and so it's really defined. This log rhythm is just to find to cancel out the exponential. So if I take log base a of a to the X, I get X. And if I raise a to the log base a X, I also just get X. So we're gonna use these properties all the time. And so it's really important to remember that these air just inverse functions of each other. Now, this is a great time to mention a very special base. So that base is E and E is just a number. It's 2.7. Give you a few of the digits 2.718 to 8, etcetera. They're infinitely more digits. It's sort of like pie. It's irrational. It just has a lot of digits, just goes on forever. Their computers right now trying toe list out as many possible digits of E. But this is just a number, and it really seems like a very random number, but it is a valid base for an exponential function. So if ffx is e to the X than the inverse is Log Base E. Fx. But we actually give this a special name. Ellen of X. So the natural lager them. So that's log base eat. Or is this number 2.71828 etcetera? Also, Sometimes you'll just see log with no subscript no base. That's understood at least some of most contact context to mean long based 10. So if you ever seen long by itself with no base, it's long based. 10. If you see this Ellen natural log, that's a long base eat so let's review a few more properties of the longer than function. So first of all, remember, let's say that now our function is log base A of X. Where is a positive number other than one If a is one, this function is not, uh, well defined. So let's see, what does this function look like? Well, recall that because we know the domain and range of the inverse function eight of the X we know the domain in range of this function. It's the domain of the longer rhythm to base. A is going to be the same as the range of the exponential base A and the range is going to be the same as the domain. Mhm. Let's not do the blackboard there. So the range of the log rhythm is going to be the same as the domain of the exponential. And if you recall that exponential functions when you graph them, are just reflections over the line y equals X. So the longer in the function should look something like, Yes, and this is actually going off to infinity is X goes to infinity. Okay. And another important thing to note is that no matter what the bases, the longer than is going to cross the X axis, It X equals one. So that's X equals one right there. So that's important to remember. So that's kind of what the shape of the longer than graph looks like. It's really going to be important later to remember what Thean behavior of the longer than is. So his ex goat gets closer and closer to zero. The longer than function is actually going to minus infinity and his X goes to infinity. The longer than is going off to infinity very, very slowly. But we'll talk a lot more about that when we get actually into the calculus of the longer than functions. So the final thing we want to talk about with logarithms right now is the basic properties, just like we talked about with exponential in All of these actually come from the corresponding property for exponential just applied using the embers instead of the function. So not really much to say here, other than we're going to use these a lot later. And so you should just not be surprised when we see logarithms and we use certain properties. And if you don't have these memorized, we'll just review, um, real quick. I would do some problems with logarithms Thio. Familiarize yourself with these properties, so logs take products to sums. So the log of a product is the sum of the logarithms logs takes exponents toe exponentially ation to products. So if I raise a number thio another number, I take the longer them that's the same issues multiplying the exponents times longer to them. This one is really important. This is called the change a base. So later on, what we're going to do is we're gonna almost exclusively use the natural law group, and there's really not a good reason why we're gonna do that other than it's going to standardize everything and the natural longer than has some really nice properties. So if we ever have, like, log based four or log based 10, what we're probably going to do is we're probably gonna immediately change to the natural log rhythm, and this is how you do that. So if I want or if I have logged based two of X and I want the natural log, then it's just natural. Log X divided by natural longer, too, and We'll see more examples of that later. And then this property just takes. Okay, so reciprocal is log. We'll just pick up a negative. And so these are the properties of the log rhythm.