Transcript for:
Trigonometric Function Properties

[Music] thank you all right welcome back again everybody uh today we're going to be doing section 5.3 we're going to talk about some properties of the trigonometric functions in the last section we got the actual trigonometric functions themselves sine cosine tangent cosecant secant and cotangent in this section we're going to talk about some of their properties and then the next section will actually review some graphing techniques before we start to look at the graphs of these functions so we haven't seen the graphs yet but we will and we'll make good sense of the graphs too but first we got to talk about some basic properties so let me get the slides pulled up there we go and let's take a look at the overview so we're going to talk about periodicity which we've actually already seen and we saw it in the last section uh intimately and we'll see it again in this section then we'll talk about some of the fundamental identities and then we'll talk about symmetry here we go so what is periodicity to start so we say that a function is periodic if there is a non-zero number P such that F of theta plus p is equal to F of theta so more specifically what we're saying is that if you take your angle Theta and you add some value to it and you end up back or I should say this where we're saying this if you take your angle Theta and you add P to it that's the same thing as just starting with Theta itself so in particular the outputs of these functions are the same the outputs of the functions right so this is the output of the function when you put in theta plus p and this is the output of the function when you put in Theta and we say that a function is periodic if those two outputs are the same for some non-zero number P non-zero number P and the number P itself is called the period of the function and we say that f is p periodic for whatever that number P is um yeah so in other words the function values repeat themselves every P units and graphically this means that the graph of f repeats every P units and so we'll see this when we look at the graphs in section 5.4 all right so we know what sine and cosine are now and because they're defined as the vertical and horizontal displacement of a point on the circle right point on the circle determined by Theta both functions are periodic and we kind of saw that when I was drawing the lines in the previous lecture right it basically goes up and down up and down left and right left and right that's periodicity that's periodicity and it's related to the fact that the thing is going around and around the circle right round and around the circle so in particular sine and cosine are 2 pi periodic because the location of the point P just repeats every two Pi radians and this is something we saw last time too so if we want to write that more specifically we say this sine of theta plus any integer multiple of 2 pi will be the same as sine of theta and cosine of theta plus any integer multiple of 2 pi will be the same as cosine of theta because these functions are 2 pi periodic that's the idea in fact all six of the trigonometric functions exhibit some periodicity precisely because they're defined in terms of sine and cosine so that's one of the reasons why I keep emphasizing that if you can grasp sine and cosine really well then you can grasp tangent and cosecant and secant and cotangent just as well you just have to make sure you're comfortable with sine and cosine okay so if you want to get more specific we can say this these are these are the uh the periods of these functions so sine and cosine and cosecant and secant are all 2 pi periodic but tangent and cotangent are Pi periodic so that's one thing to take note of um the short answer is that tangent has a period of Pi and so that's why cotangent does as well we'll see more why that's true when we start to look at the graphs but algebraically it's this if you do any integer multiple of Pi and you plug that in I'm sorry um you add that to Theta you're just going to get the same tangent value at that point there you go all right so this one we actually had practice with this on the last section so let's try to find the exact value of sine 17 pi over 4 cosine 5 pi and tangent 5 pi over 4. so for this one now that we've done it a couple times in the last section I want you to try to do it on your own first and then we'll do it together a little more quickly than last time all right so think about it and try it on your own right now all right now let's try to do it together here we go just like we did in the previous section we're going to mod out by 2 pi so first let's do this one okay we're looking at the angle 17 pi over 4. which is the same as 17 4 of Pi let's see 17 divided by 4 is 4 and a quarter pies okay so what do I do draw my unit circle I'm going to go around four times so one two three four oh sorry I should say around 4 times pi right so 1 Pi 2 pi 3 Pi 4 pi and then a quarter pi so we're right there just like last time this is going to be a 45 45 90 triangle so we get that sine of 17 pi over 4. is equal to root 2 over 2. done not too bad right so hopefully you're getting a little quicker at these now because you realize you just got to do division and then count just appeal to Counting when you can right all right so let's do the next one cosine cosine of 5 Pi well 5 Pi we're going to go around 5 times but it's just a it's just an integer multiple of Pi so we're going to land at either Pi or we're going to land at either Pi or 2 pi one of the two let's see so one two three four five there we go so we're looking like this oops all right what's cosine of this angle this is the angle pi cosine of pi is negative one all right horizontal displacement is negative one right what about tangent of 5 pi over four place this right here 5 pi over four same kind of thing um 5 pi over four that's one and a quarter pies one and a quarter pies so you're gonna go a full pi and then a quarter pi and you'll end up down here this is going to be a 45 45 90 triangle root two over two root two over two but now because it's in the third quadrant there these are going to be negative value now we know what sine and cosine are we can find tangent we get uh maybe I'll cut through it a little bit we're going to get this tangent of 5 pi over 4 is going to be negative root 2 over 2 over negative root 2 over 2 which is 1. and then it's always good to do a little check right what am I saying when I write this down what does that really mean tangent of 5 pi over 4 is 1. that means the slope of that line is one does that make sense when we look at the picture think about it for a minute and see if it makes sense that we would get a slope of one yeah that's right it does right when you look at this line that Line's got a slope of one this is actually just the line Y equals X the 45 degree line the pi over 4 line now it all makes sense okay uh this is just all the details so if you want to read through this on your own you can all the details of what we just did now let's talk about some identities so uh well actually one thing I'll mention is that this is going to be a much quicker lecture because we got the big ideas in the last one um but there are some fundamental identities that we're going to start with there are actually dozens and dozens of these identities that we're going to learn throughout the class but there are only a handful that you should memorize there's only a handful you should memorize and I'll be very very clear about which ones you should memorize the rest of them I'll just give them to you if you're if you're taking the test because there's just so many of these identities just so many and there's no way to memorize them all comfortably and there's no way to keep them all in order in your head comfortably but there is a handful of them in particular I'd say six or seven six seven or eight maybe that you want to have memorized and I'll emphasize them as we go along but this is one of them this is one of the identities that you really really do want to memorize this is called the Pythagorean identity the Pythagorean identity and maybe you can kind of see why this says cosine squared theta plus sine squared theta equals one so one thing I realize I mentioned it on a later slide but not this one do I mention it here yes okay there we go so let me cut to this slide really quick um it's convention to write sine Theta squared as sine squared theta and the same thing for cosine and tangent cosecant secant cotangent Etc um the reason is purely notational it's purely for Elegance it's just because it avoids cluttering up Expressions a lot because when you start to have a ton of parentheses flying around which we will it's nice to not have to include all of them so don't let this confuse you or trip you up but quite literally if you see sine squared theta it means sine of theta squared if you see cosine squared theta it means cosine Theta squared if you see tan squared theta it means tangent of theta squared and so on that's really all it is it's just a notational choice um and yeah like I say here in the second point this identity is called the Pythagorean identity precisely because x squared plus y squared equals 1 is the Pythagorean theorem that's how we got this result that's the equation of the circle which it comes from the Pythagorean theorem which comes from those right triangles so it's all intimately connected it's all intimately connected as long as you can remember the big picture of the unit circle and those special right triangles that's it all right so let's prove this identity the proof is very straightforward the proof is like one or two lines there you go so any point x comma y on the circle we know satisfies this equation by definition X I'm Sorry by definition cosine is the horizontal displacement and sine is the vertical displacement so we can quite literally just substitute in X and Y or I should say substitute in cosine and sine for X and Y respectively and we get this that's it there you go that's the proof all done um let's see another one so this is another identity and this is the one that I call I tan in a second and the reason I call it that is because one of my previous students years ago told me this mnemonic for remembering this identity because I I had never heard it before and he blurted it out and I just I fell in love with it immediately it's a great mnemonic to remember this identity the mnemonic is I tan in a second and what's the identity it's 1 plus tangent squared theta equals secant squared theta I tan in a second you just got to remember to square the tangent and the secant okay now let's see the proof the proof is also remarkably quick uh we're going to start with the Pythagorean identity there we go and we're just going to divide both sides by cosine squared divide both sides by cosine squared there we go uh cosine squared over cosine squared is one sine squared over cosine squared that's just tangent that's tangent squared right sine over cosine is tangent so this becomes tangent squared and then 1 over cosine squared that's just secant squared by definition right what what secant secant is 1 over cosine so 1 over cosine squared is secant squared and there we go that's the proof all right then the last one this one is icotan and a cosecond so just like we had I-10 in a second for the last identity it turns out there's another one that's related it's one plus cotangent squared equals cosecant squared there you go uh what's the proof the proof is also remarkably quick you start with the Pythagorean identity again there you go divide both sides by sine squared instead so you divide both sides by sine squared well like we had seen before oops cosine Theta over sine Theta that's cotangent so this is cotangent squared sine squared over sine squared is one and one over sine squared well 1 over sine is cosecant so 1 over sine squared is cosecant squared and there we go Tada that's it I cotan and a cosecond all right so these are three identities that you must memorize you really really need to memorize these ones you will use them extensively in calc one you'll use them in calc 2. actually I'll say you'll use them in calc one you will use them a ton in calc 2. so this is my my warning to you from the past to your future basically saying memorize these identities get to know them now make sense of them now get comfortable with them now because then when you're in calc 1 and calc 2 you'll be like oh I got this you won't have to like memorize them then and relearn about them then all right so there you go learn about them right now okay let's do an example let's find the exact value of this expression oof this expression looks kind of like a beast um take a moment look at this expression see if you can make sense of it and then see if you can find the exact value then we'll do it together all right now try to write something down see if you can make sense of something written there and get it started all right now let's try to do it together um for one since we've just seen this new notation it might help to write it in notation that's a little more familiar so if you need to do this step please do it it's not wrong to do it but sometimes when stuff gets a little bit funky you have to use parentheses because parentheses are your friends and parentheses can help you keep things grouped the way that you want them to so when I look at this I see this as sine of pi over 12. squared sine of pi over 12 squared and then here I see this as 1 over secant pi over 12. squared okay so now maybe you can see actually see this is one of the reasons we tend not to include all of these extra parentheses let me do this here instead um pi over 12 parentheses squared this is why we like to use that more cleaner notation but in this case it might make things a little bit easier to see so we've got sine of pi over 12 squared and then we have 1 over secant squared but what is secant I remember secant is 1 over cosine so this is 1 over 1 over cosine pi over 12 squared so this is going to be sine of pi over 12 squared plus cosine the pi over 12. squared hmm but check this out this is sine of an angle squared plus cosine of an angle squared right that's the Pythagorean identity this is one this is equal to one because you've got sine squared plus cosine squared so the answer is one that's what it is the value is 1. by the Pythagorean identity and that's it that's it so this is a lot of what you'll see in this course with these identities it's going to be a lot of manipulating these expressions and finding equivalent forms of them that may or may not be simpler so this is an example of what you'll see a lot more later in the course okay now let's try this one so think about this one on your own and then we'll try it together so start by pondering it to yourself maybe say some things out loud and then try to work on it you're on your own okay all right now one thing I want to remind you is try to say some stuff out loud too I realize I haven't been throwing this up enough I've been going straight to like the writing part but saying things out loud even if you're alone can really help your brain process stuff just talk to yourself or you can talk to your dog or the wall or your cat or your friend but saying stuff out loud can help okay now let's try it together let's see if we can make sense of it it says given that sine of theta is one-third and cosine Theta is zero find the exact value of the remaining trigonometric functions hmm okay so it doesn't even mention that we're using a unit circle or anything like that so we just know that sine is one-third all right sine is one-third well what did we know we remember in the last bit the last section we knew that sine of theta was equal to Y over r right so if you had a circle that wasn't necessarily radius one it would be y over R vertical displacement divided by the radius so what we know is that this is also equal to one-third so now we can just do a direct comparison if y over R is equal to one over three this tells us that Y is equal to one and R is equal to three there we go okay so now we know what R is we can figure out what cosine is because what's cosine cosine Theta was X over r right X over R hmm but then how are we going to find X that's a good question let me start with this okay we're going to have X over 3 but what is X gonna be hmm how could we do it what do you think right the Pythagorean identity right so we know what Y is we know what R is we just need to solve for x and we also know that x squared plus y squared equals r squared so kind of filling this in we're going to get x squared equals one oops sorry plus 1 squared equals 3 squared so x squared plus one equals nine so x equals the plus or minus square root of eight and then maybe I'll simplify that a bit x equals plus or minus 2 root 2. just like so and there we go so now we just need to know if it's the negative or the positive one remember whenever you take a square root you must include a plus minus then you can toss out a case based on the situation so here how do we know which one to pick what do you think it's going to be the negative one that's right how do we know that we're given that cosine of theta is less than zero but we know that cosine is basically the horizontal displacement scaled by the radius right so we know immediately that we're going to have to pick the negative one we're going to have to pick the negative 1 because cosine is less than zero so that tells us that X oops let me just do this X is going to be negative root 2 over 2. negative root 2 over 2. okay and that's all we needed so wait if x is negative root 2 over 2 then that means that cosine is what negative root 2 oops negative 2 root 2 over 3. yes all right so now we've got sine we've got cosine we can find tangent we can find cosecant we can find secant we could find cotangent we can find all of them all we needed to start was sine and cosine so here we go oh this is me just saying what I did there you go Pythagorean theorem or the Pythagorean identity there we go so we had this this was given this is what we concluded with our calculations once we have those two values we can find tangent just do y over X 1 over negative 2 root 2 to find cosecant do R over y to find secant do R over X and Define cotangent do X over y there we go and that's it that's the whole idea so once you get sine and cosine everything really does just kind of fall right out all right so that's it for identities for now the next thing we're going to talk about is even evenness and oddness so there are these functions that are called even and odd functions it's a very specific property that can be very very helpful depending on the context so we say this a function is even if and only if f of negative theta equals F of theta for every input Theta let me emphasize this right here so intuitively what this means is if you change the sign of the input it does not affect the output at all that's what it's saying so if you change Theta to a negative angle instead that's not going to change the output of the function that's what it means to be even an even function has that property a function is odd if it does have an influence when you change the sign of the input so a function is odd if f of negative theta equals negative f of theta so what this says is hey if you change the sign of the input it changes the sign of the output that's an odd function we say that's an odd function um graphically do I say it on this slide I think I waited because we'll talk about that in the next section let me double check yeah so we can see this on the graph as well and we'll see it in section 5.4 but even functions are symmetric about the y-axis as it turns out even functions are symmetric about the y-axis odd functions are symmetric about the origin for example but we'll explore that later in the next section for now the big idea is that if you change the sign of the input does that change the sign of the output or not if it doesn't change the sign of the output at all you've got an even function if it makes the sign of the output the opposite you've got an odd function so all together for our trigonometric functions we get this cosine and secant are even functions and then the rest of them are odd functions there we go so when I say cosine and secant yeah so odd function odd function odd function odd function even function even function and we're going to prove one of these but if you want to make sense of why this is happening guess what you're going to use think about it for a minute what do you think I'm going to say that's right the unit circle if you want to prove any of these things you start by just looking at the unit circle looking at the definition of sine and cosine and these results fall out immediately so let's check it out um here's the proof that sorry I wanted to say we're going to prove this one we're going to prove that cosine is in even function so what that means is if I change the angle to a negative angle that just doesn't change the horizontal displacement but even when I like say that out loud it's kind of that that's it right that is the proof that is the proof and here's the picture of it there we go that's the picture of it right so if I change my angle from a positive value to a negative value that doesn't change the horizontal displacement so there you go by definition that means that cosine negative Theta is equal to cosine Theta or Not by definition by the picture and then now by definition this means cosine is an even function there you go pretty nifty huh not too bad um why is this helpful well evenness and oddness comes up in a lot of contexts and it can cut your work in half or it can cut your work to zero entirely you'll see it a lot in calc 1 and calc 2 actually so do get comfortable with even and odd functions but let's try an example here a simple example see if you can do this using even in oddness properties instead of just using only the unit circle so think about it for a minute now try to say something out loud to yourself anything at all about what you're seeing here what you're reading and how you would approach the problem all right now try to work it out yourself see if you can use even an oddness to find the answer to these these things these values okay now let's do it together so if I'm going to use even an oddness I remember that sine is an odd function what does an odd function mean it means this sine of negative 45 degrees is going to be negative sine of 45 degrees because if I change the sign of the input it changes the sign of the output all right but what sine of 45 degrees unit circle 45 degrees 45 90 triangle it's root 2 over 2 so this becomes negative root 2 over 2. done there we go so now we didn't even need to contemplate going uh in Reverse right although we could you could easily just look at negative 45 degrees and go that way and get the same result there's more than one way to do it okay what about cosine of negative pi that one cosine is an even function so cosine of negative pi is the same as cosine of pi but that we know is just going to be negative one right over there you'll have negative one okay now tangent negative 37 pi over 4 let's see tangent is an odd function so we know that we're going to have negative tangent 37 pi over 4. right but then what is 37 pi over 4 look like we got a mod out by 2 pi think of it as multiple of Pi and get rid of all the the repetitive get rid of all the repeated rotations all the repeated revolutions so let me do some side work here 37 over 4 Pi see 37 divided by 4 is how much see 36 so 9 and a quarter pies nine and a quarter pies so on the unit circle I'm going to go around and around for nine pies and then another quarter pi so because it's 9 we're going to do um uh we're gonna do a a bunch of full rotations and then another pi and then a quarter Pi so like you'll go one two three four five six seven eight nine and then another quarter so you'll be down here all right that's uh what's the tangent of that line oh I can see it in the picture right the tangent of that line what's the slope of that line the slope oops the slope is going to be one the slope is going to be one so this is going to become negative one there you go all right boom there we go negative tangent pi over 4. negative one and that's it so that's how even an oddness can help you for evaluating these functions um but what's what's nice about this is if you're comfortable with the way that we've been evaluating these functions using the unit circle even an oddness doesn't save you too much work but what you're going to find out when you get to calc 1 and calc 2 is that it literally will cut your work in half or cut it down to nothing because of the way that these functions behave when you integrate them over a symmetric interval but that's for a later class so now you can go play try these out on the homework ask questions if you have any but that's it for this section so I'll say thank you again for listening I'm glad this one wasn't as long and I'll see you next time thank you