Basic Properties of Trigonometric Functions (Precalculus - Trigonometry 8)

welcome to another video in this video we're going to talk about some of the basic properties of trig functions sine cosine tangent cosecant secant and cotangent just as as they relate to each other start building some of these ideas called identities and toward the end of this video we're going to start talking about that how identities let us discover a lot more about trigonometry than we would know just by given one trig function so we can we can extrapolate from that or infer from that based on these these identities that we're going to start unpacking right now so we're going to take a look at a unit circle very quick talk about what sine and cosecant how they relate to each other in terms of the x and y coordinates of the points on the unit circle same thing with cosine and secant same thing with tangent and cotangent so let's get started when we talk about sine of some sort of an angle that's swept out counter-clockwise positive on a unit circle or another circle for that matter but unit circles is very convenient for us is when we when we correlate these things with the sign of an angle and a point that's on a unit circle we understand hopefully because i explained a lot that sine of whatever angle this is is going to be given by the y-coordinate of whatever that point is so in our case we're going to have sine of theta equals y remember what trigonometry does trigonometry relates an angle to actually a a ratio of y over one or the opposite over the hypotenuse and the hypotenuse is 1 because we're in a unit circle of a right triangle and so we're always doing that we can't just say sine equals doesn't make sense but sine of some angle is going to equal the y coordinate of a point on the unit circle well because cosecant is this this idea of a reciprocal relationship to sine we can define cosecant of theta as one over that y notice what this is doing because sine theta is y over one i know that one's not there but that one is that radius right there y over one if you reciprocate it is one over y or instead of looking at this if you remember opposite over hypotenuse cosecant would be hypotenuse over opposite for whatever angle that is this is going to play in pretty big when we start talking about domain because while y we're going to get all real numbers we're letting the bag here cosecant we're not because what's going to happen is when y equals 0 that becomes a problem do you remember that i hope you remember that how domain when we have fractions it's an issue when our denominator equals zero we get some undefined values so we have some domain issues things that we cannot plug in and we're going to get that here how about cosine well cosine theta when we take a look at this cosine we've associated with the adjacent or the x side of this so when we take a look at our unit circle cosine of whatever angle we're sweeping out is going to be the x coordinate we can think of this as adjacent over hypotenuse that would be the adjacent side or x side over the hypotenuse of one that would give us just x because secant is defined as the reciprocal of that cosine we can put 1 over x here so instead of adjacent over hypotenuse it's hypotenuse over adjacent that's 1 over x i know that i'm relating this back to the right triangle i'm doing that on purpose because we're going to get there it's a really long time from now but i don't want these two things to be separate i wanted to i want to understand that whether we're talking about sine cosine tangent this these trig functions on a unit circle or a right triangle we're talking about the same thing we can relate them the same way the only one of the main issue differences here is that we can also talk about arc length as it relates to a unit circle but in general i want you to understand that how sine is associated with the y output or the y coordinate but it's also associated with opposite over hypotenuse that same thing for cosine tangent and all the reciprocal functions we can relate them both ways now for tangent tangent is this idea on a right triangle of the opposite side over the adjacent side of a right triangle but for us that would be a y over x we can also get sine over cosine and likewise for cotangent we're going to reciprocate that and get x over y we can think about that a right triangle as adjacent over opposite those are really the the major relationships that we have when we're thinking about trigonometry on a unit circle now we're going to get to inverse functions eventually as well where we're going to be using a unit circle in a slightly different way to go backwards but as it stands right now i really need to get that i need you to understand that sine cosine tangent have reciprocals cosecant secant and cotangent respectively i need to understand that sine of theta relates an angle to the y coordinate cosine is an angle to the x coordinate tangent is an angle to the ratio of those two things and that's really important so now we're going to transition just a bit we're going to talk about domain and range of these trig functions and we're going to do it just by looking at what the outputs could possibly be so let's start with sine remember when we're looking at domain we're trying to think what can i plug in that gives us an actual output not just plug in anything you want what can i plug in or evaluate into my function that's actually giving me something and so really what we're looking at is what angles give us some realistic outputs here that's how we're going to start to find our domain we're going to start with sine and work our way through this we'll do all of the domains and ranges of each of these functions as we go because they interplay off each other a lot especially down to tangent and cotangent so just just remember with domain that one issue we have is that if you ever try to divide by zero or something gives you an output that is divided by zero that input's not valid i'm going to repeat that a slightly different way what we're looking for is to exclude any angle that's our inputs remember talk about domain here exclude any angle that will give us an undefined relationship something over zero so let's take a look at that for sine if we think about sine and if sine is the y coordinate that's why i kind of pounded that into your head just right now if sine of theta is giving you the y coordinate of whatever this points are in the unit circle well let's let's look at y the y coordinate starts at zero so for this angle you start at zero and it's going to get larger larger larger larger larger until it peaks at one one is the largest value that we can get for y and it starts dropping again dropping dropping dropping drop remember i'm focused just on the y coordinates just sign i'm gonna show you slightly different ways a second and then it starts dropping dropping dropping y starts becoming negative here until you get down to negative one that's the lowest it ever gets and it starts climbing again still negative but it's getting closer and closer to zero and then it starts to repeat so for our y values are there any y values that are undefined well what i'm asking is is there anything that i have an angle where my y value is like i don't know or i get something over zero well no that's not true i'm going to show you slightly different way i'm going to use my left hand on the y axis and my right hand to go around the circle and check this out as we climb the y value just gets higher on the switch it gets lower it bottoms out at negative one and then it comes back to zero so all my finger did all my y axis finger did is it moved from here to here then from here to here back from here to here and then back from here to here when we get to graphing i'm trying to make the connection right now when we get to graphing sine is going to do exactly that it's going to climb fall climb from zero to one back to zero to negative one back to zero that's what the sine function is going to look like just this nice repeating curve what do you mean repeating well as soon as you make one full rotation this two pi you're going to get back to where you started and you're going to start getting the same exact y values over and over again that's called a period we're going to get there in just a little bit so if i if we're looking at this think about your angles right now are there any angles that give you this undefined output are there any angles that give you a y that's not on here well no the only y's that we're getting out are from 0 to 1 and then from 0 to negative 1 back and forth between that so all of our angles all of our thetas are actually defined we can plug in any angle and it's going to give us this actual y value it just goes from 1 to negative 1 or negative 1 to 1 that's it so what we're going to say is that for sine we're going to get all real numbers there's not an angle that you can't plug in that's not going to give you something in terms of sine so that y is always defined for us it's always between negative one and one and that's exactly we're going to get for our range right now so you can plug in any angle that you want notice that anything you want anywhere on the circle as many times around as you want to go but ultimately your your y-axis finger is just going to go from here to here somewhere between there well how many numbers are between negative one and one an infinite number of them but you can't get outside of that well that's the range the interval of outputs that you can get is called your range now you can plug in anything sure and get out something between negative one and one so our domain is theta can be all real numbers anything that you want to as many times around as you want to go positive or negative but you can only get out negative one to one now there's no hidden missing numbers in there there's it's continuous here from negative one to one as far as our range is concerned but we don't have anything undefined we don't have anything outside that so our range is you can get out negative one all the way to one we're going to use a bracket to represent that so so we can show two different ways we can show interval notation negative one one or we can show this with inequality notation here negative one bracket and one bracket the sine theta that's that's the largest indicator remember this is actually y and so we can see that y never gets less than negative one never it's bigger than one i know i spend a lot of time on sine but the rest of these are going to flow from that so that or at least the idea let's look at cosine cosine is just x remember cosine is the cosine of whatever angle you have is mapped to the x coordinate let's look at x the same way we did y if i start here with an angle of zero radians or degrees and i start going positively around my circle our x starts at one this is the point one zero as we start going my x gets smaller until it gets a zero then it starts getting negative until it gets to negative one that gets pot of well grows again until it gets back to zero and then positive until i get to where i started so for the same exact reason that sine has a domain of all real numbers cosine also does you can plug in any angle and it's going to give you some sort of an x value not nothing that's undefined nothing that's outside of the range of negative 1 to 1. and so these look very similar so cosine of theta the domain for theta is all real numbers whether you're talking about sine or whether you're talking about cosine remember that both of these things relate an angle to some sort of an output x for cosine and y for sine our range is also the same you can get out for cosine negative one to one anywhere along that interval of the x-axis remember we're just looking at x right now i want to point out a few things just some common misconceptions that students who are just seeing trigonometry the first time or really have never understood it make the first one is that while sine and cosine have the same range and they have the same domain they're talking about different things remember trigonometry really relates three pieces of information it relates an angle and a ratio of two sides or an angle and some way of relating these these coordinates of a point and so what we're talking about here is is really a different type of domain i know that you're used to a domain being an x-axis idea that that's true for for all of our functions that are defined in terms of x and our outputs in terms of y but really we have one type of input which is a domain of an angle and so when we're talking about domain we're talking about what angle can this actually get well sine you can give me any angle why because our outputs are one of two different types of outputs um a y or an x or a comparison of them and so our inputs are angle ideas not an x idea and that can be confusing for students that's the second thing that people get confused on is to say well i thought my domain was just negative one to one and it's not that's actually your range so the first thing is that sine and cosine are different they're not the same thing the second thing is that when we start talking about domain and range it's a little different from normal our domain is what angle can you plug in any angle will give you out some sort of a real y value for sine and some sort of a real x value for cosine i hope you're seeing that i hope you're seeing that the domain is is different than normal it's not just an interval with x it's an interval of an angle so an interval of where you can go on a circle how what radians can you get or degrees can you plug in and get something out for sine for sine and cosines any angle will give you a y or an x respectively what can you get out for those y's x well well now the x axis gives you this interval for cosine this range for cosine and the y axis gives you an interval or this range for sine and that's an idea it's completely different right and so our domain is yeah you plug in any angle but for sine you can't get out anything besides negative one to one or some value between them for cosine it's an x's x axis idea you can't get out anything from negative one to one or some number inside of that um along the x and along the y for cosine sine respectively and i hope i'm making that that makes sense to you um the last thing i want to talk about is graphically notice how sine when i start at zero sine starts at zero gets larger smaller to zero negative and then back to zero cosine does that but shifted and we're going to get an identity later that says that but cosine starts at 1 so cosine is going to start at 1 drop to 0 go negative and get back to 1. so it's a different looking graph they're very very similar in how they're they're shaped and in fact you there's identity transfer one to the other to translate a little bit to shift it uh but for now i need to understand they are different even though they look very similar because one is a y idea and one's an x idea okay i've talked a lot about that let's talk about tangent now tangent's different you see tangent takes out a relationship it says what angle can you plug in such that i get a real defined output out well we know something about domain we know that we can't plug in numbers that cause us to get some sort of fraction with zero on the denominator out and so we're really going to start taking a look at that first i'm actually going to define the range first and then the domain why well because the range is x's y's the the angle is what angle caused or that sorry the domain is what angle causes that that output so we're going to take a look at some outputs for tangent that we really shouldn't allow remember that tangent is y over x and so we're going to look for this as all right if it's y over x then our x where our x is 0 we have a problem so let's start taking a look at our unit circle here our x is one that's no problem remember we're looking at tangent as this y over x so let's see zero over one that's fine and we're going to get positive number over positive number that's not a problem until we get to right here at that point we have the first instance where tangent would have a major issue tangent is one over zero here remember if x is zero this is a problem our x starts from one and as we're going around it goes to zero here negative and then back to zero so at the first and third quadrant angles at pi over two and at three pi over two we have some major issues and so we've sort of taken a look at the range when we're thinking about this thinking what outputs are wrong and then eliminate those inputs that angle from the domain of tangent in this case so we said hey look x is 0 here that means that pi over 2 we're going to get something that's undefined well what that means is that input that pi over 2 is not allowed for tangent same thing happens as we go all the way to here 3 pi over 2 that means that 3 pi over 2 that input is not allowed because it would give us negative 1 over 0 that's something over 0 that's undefined so we've taken a look at range and now we're going to define our domain because of that and so that that's pi over 2 90 degrees 3 pi over 2 270 degrees 5 pi over 2 and you can see that we can go around the circle positively or even negatively so like negative pi over 2 and have some domain issues with tangent we have a slightly easier way to write that in a little bit but these are this is just your idea so every pi over 2 that you get sorry pi over 2 or pi over 2 plus or minus pi that we get we have an issue 90 degrees 270 or another 180 degrees past that or subtracted from that we're gonna have some major problems here i hope you understand that i hope you understand that there's certain angles of tangent because x can take the value of zero right there that we're gonna have some issues with now what was our range well this is this is quite interesting for tangent because tangent takes two values y and x and puts them in ratio y over x x can get very very small when y is close to one well what that means is you get some very very large numbers and say that again when y is close to one x is very close to zero so what's a number very close to one divided by a number very very close to zero it's getting close to infinity so as we approach this pi over two we're getting close to infinity that's why we can't even plug in one over pi over two because we get one over zero that's taking one divided by zero you can divide one by zero it's undefined but just to the left and right of that we're close to positive or negative infinity we're going to see that graphically in just a little while so our range is you can get out anything for tangent now that we've dealt with these we can deal with the reciprocal functions quite nicely as long as you understand this idea the idea of why these can't happen is because of those inputs would give us an undefined output that's a problem so that that means we can't allow that in our domain if you understand that hey taking a ratio of two real numbers on an interval of one to negative one or negative one to one for both these cases you can get anything out of that if you divide them in uh in those relationships that i've shown you like hey y is getting real close to one and it's getting real close to zero one divided by something real close to zero is getting to infinity well if you understand that then we're going to get some very similar things here i want you to think about cosecant so cosecant of theta says what you're going to do is you take 1 over y well wait a minute this is going to have problems if we allow some angles that let y equal 0. so all we've got to do is look at our unit circle and realize that hey if sine gives us y and cosecant is 1 over that y then where y equals 0 we have issues other places are going to be fine but here at this angle 0 we're going to have a problem because this is going to try to take 1 over zero and then oh well that's positive y all the way from here to here that's going to try to take one over zero this is negative y from here to here that's fine so we're going to have two values on the first revolution around the circle they're not allowed at zero and at pi and and so on and so on and if we talk about degrees that'd be zero degrees and 180 degrees and 360 degrees and every 180 degrees past that so we can go round and round around and every so often at every one of those times where our y is equal to zero we'll have something that an angle that we can't plug in that we can't evaluate that is giving us an undefined output and so what this means is that is that in terms of our range we can get out something kind of strange you see this is this is kind of interesting but if we're going to start dividing 1 by y remember that's what this is right this is 1 over y well y's largest value is one and y is large the smallest value is negative one so if we divide one by one i'm sure we're gonna get one or one by negative one we're gonna get negative one but one divided by something between there remember this is between negative one and one well those all those values are smaller than one one to a one divided by something that's mostly smaller than one is going to give you values larger than one what in fact the closer you get to zero the larger these numbers are that doesn't change that oscillates between negative one and one so one divided by negative one is negative one that's that's actually one of our bounds of this of our range and one divided by one is 1 but 1 divided by let's say 0.5 well that's 2 or negative 0.5 that's that's that's negative 2 or let's say 1 divided by 0.1 well that's 10. these numbers can get really big 1 divided by .00000 can it be there yeah we can have that here's why it's very very small one divided by very small numbers is very large or very small negative numbers is very large negatively if you will so what our range becomes is we can get out a lot of negatives up to negative one and then one to infinity so the only things we can't get out is this interval between negative one and one it it should make some sense that i'm basically reciprocating everything right so we're going to get this this outside idea of negative one to one sure we can get out negative one and one just by dividing but remember that y is bound by that y is it can only be smallest value negative one largest value one everything else is between there dividing one by some numbers smaller than one you're going to get values larger than one or out or outside that that range of negative one to one now how about secant the same exact thing happens here you can probably see it it's just that our domain is going to be different because we're looking for where our secant is one over zero where x is zero so where is x zero x is zero at pi over two and three pi over two okay now why in the world would this have the same domain as tangent does why well this says if i take y divided by x where x is not matter where x is 0 that's given us issues and this is saying the same thing look at our denominator here's a denominator of x here's a denominator of x y is defined everywhere one is a constant so the only issues we could possibly have are where x can equal zero they're the same exact angles where x can equal zero remember our inputs are angles here where our angles are pi over two three over three pi over two five pi over two and etc you're going to get x values of zero that makes these inputs give us an undefined output for both of those cases now our our range is going to be the same as for cosecant but remember we're talking about a different a different output this is 1 over y and this is one over x so he's talking about x values okay last one then we're going to talk about the periods of all of these these trig functions uh period is how often they repeat so we'll talk about that a little bit how about cotangent well for the same exact reason that uh secant matches up with tangent cotangent is going to match up with cosecant what does that mean cosecant says if your y equals zero you got problems well same thing with cotangent if your y equals zero you got problems here so cotangent takes a relationship of the x divided by the y so any angle that gives us an output where y is equal to zero is going to give us a problem something that's that's that input is undefined and so what we're going to say is that r theta cannot be the same exact values that are cosecant is cotangent is creating this relationship of an angle to the x coordinate divided by the y cosecant is creating this relationship between angle and one over y so where this equals zero is the same place where this equals zero and both of those would be undefined so we're just going to basically copy down those so at zero at zero your y is zero and at high your y is zero and at two pi or why is it zero and lots of other places as we keep going around and around that unit circle both positively and negatively for the same exact reasons that tangent of theta can give you an output of all real numbers anywhere y divided by x same thing happens here x divided by y well x is all real numbers and y is all real numbers the only things that we're excluding are where your y is 0 you can get everything out of that as well i probably could have just given all this to you and said hey here's your domain but i hope that you stick with me and that you walked through it and understand why why it is you can plug in any angle here hopefully you understand a little bit more that what trigonometry does is relates an angle that is what you're plugging in here that's what you're evaluating and your outputs are something on the x-axis or something on the y-axis or ratio of those two things hopefully you understand that now so our domain is talking about what angles can you plug in any of them because your y is defined between negative one and one everywhere for cosine any of them because your x is defined negative one to one everywhere on the x axis tangent you can't plug in things where where your x is zero well that's going to be these particular angles you can get out of everything because of that ratio cosecant you can't plug in things where y equals zero any angles where y is going to be zero why well because it's based on your sign and your coseting can't look at anything where x is any angle where x is going to be zero and that's why we exclude those cotangent you can't plug in anything where y is equal to zero again because that's the reciprocal of tangent so hopefully you understand a little bit more about where that's coming from we'll talk about what our periods are in just a second when we hear about what a period is in trigonometry it's an interval on on which a repeatable interval on which your outputs of that trig function repeat and so that idea of being a repeatable interval on which your trig functions output or repeats what that means is that we can we can really repeat a lot of the outputs every so often and that period is going to really help us in graphing really understand what these trig functions are all about so let's take a look at what this period means this interval of repetition for your tree function basically let's take a look at sine so what we're looking for is where is this repeatable repetition of your output happening well here's our outputs our sign is in blue here so here's zero here's one half root two over two root three over two and one and i want to show you something on how to how to really think of this if you want to memorize your unit circle really quickly here it is sine always counts up and cosine counts down from where you start so so check this out here's root 3 over 2 root 2 over 2 root one over two and root zero over two this is actually root four over two so if you think about that this is the square root of four over two that's one square root of three over two square root of two over two square root of one over two square root of zero over two so you can count down with cosine cosine count on the side counts up square root of 0 over 2 square root of 1 over 2 square root of 2 over 2 square root of 3 over 2 square root of 4 over 2 and then counts back down it's a nice way to remember that but let's let's look at let's just sign out but let's look at the outputs of outputs as it relates to our angles which are our sine outputs so zero one half root two or two we get our output starting to repeat right here you go oh then our our period is a repetition from here to here well i suppose but it's not repeatable because this angular distance from here to here is not the same as from here to here even though those are the same that angle is pi over 4 and 3 pi over 4 or 45 degrees and 135 degrees that's not the same distance or angle movement if you will as it was from an output of one half to one half so we're not going to consider that to be a period what we want is this every single output is repeated on this particular interval of an angle well what that's what's going to happen is we don't get another one half past this one until we get all the way back on one full revolution or two pi the period of sine is two pi well for cosine we have exactly the same thing we get 3 pi over 2 and the next time we see 3 3 pi over 2 is all the way back right here well that's great but the other in the other outputs of x or cosine theta they're not repeated in the same interval as that so this root 2 over 2 that's repeated a lot sooner than that root 3 over 2 was so what we're looking for is this this interval this angular interval on which all of our outputs are repeated that often for x and that also is 2 pi i'm going to explain that here right now so we don't get back to the square root of 2 over 2 and one half until we make one full revolution that's the same exact interval on which we're getting those outputs back so sine and cosine both have the same what we call period this same interval on which we're going to repeat start repeating all of our outputs for both y and x now tangent's weird we might think well tangent is going to be 2 pi also but it's not it's not because what we're looking for are this interval on which all of our outputs are starting to be repeated well for tangent that's sooner and it has to do with the signs this positive divided by negative negative divided by negative is given us a positive so check it out on pi over 4 we get square root of 2 over 2 divided by square root of 2 over 2 that's what tangent would be tangent is y divided by x we can see that right here sine of cosine or y divided by x so that's going to give us one but right here we also get one negative square root of two over two divided by negative square root of two over two that gives us positive one so for tangent we can see here pi over three and four pi over three we have the same relationship these are both positive these are both negative but when we divide them they give us the same output and so because we relate tangent as y over x and because positive divided by positive and negative divided by negative are both positive positive divided by negative and negative divided by positive are both negative we're going to get a period or this interval on which all of our outputs for our trig function tangent repeat is pi not 2 pi now what's very nice is that cosecant and secant because they relate to sine and cosine respectively we have the same periods and the same thing for cotangent as it relates to tangent so what this does is this gives us what is called an identity what an identity is is this truth about trigonometry that works all the time and allows you to extrapolate information from it so here's our very first identities well kind of third ones i didn't call them identities but that's one of them tangent equals sine of cosine as far as our angles are concerned so what these do is it says okay here's what we can do we know that because our output sign repeats for every one of our outputs not just some of them some of the times but every one of them all the time on a period of two pi a repetition of an angle two pi that's one full revolution or another full revolution because that happens we have this identity we have this identity that hey if you have sine of some sort of an angle then if you add 2 pi or subtract 2 pi or multiples of that what's that k that k is a whole number like one two three four five if you add multiples of two pi so add two pi or four pi over six pi or eight pi or subtract two pi four pi six by eight pi any any product of that two pi well then you're going to get the same exact output really get this it's going to make the rest of this make sense really get this that sine of this theta this angle is a y value now go a full rotation you're the same y value go another full rotation you're at the same y value doing that a lot of times or negative a lot of times you're just getting the same y value all this says is sine your your y value basically sine of your angle plus or minus full rotations is the same y value as whatever that angle is so ultimately you can add or subtract multiples of 2 pi and get the same output as it relates to sine the same thing happens for cosine because cosine is your x value of whatever angle you have that point on the unit circle so it says hey cosine of whatever angle you're going to get an x value if you go full rotation you get the same x value either way you go positive or negative you're going to get that same thing that's what this is any multiple of full revolutions is going to give you the same exact x value you can see all the time so if you went to let's say pi over 6 your x value your cosine of pi over 6 is 3 pi over 2. go full rotation you're back there go again you're back there go backwards a couple times you have the same x value same output for tangent it's quicker than that so for tangent because of the way that our positive and negative interact positive and positive is the same as a negative divided by negative because of that our period was sooner and we have a slightly different identity it says that if you start adding pi or subtracting pi from your angle that tangent plus or minus some sort of a multiple of pi 1 pi 2 pi 3 pi 4 pi whatever you're going to get the same exact output as tangent of whatever this angle is plus or minus those multiples of pi remember what that k is is just multiplying pi by simple numbers so 1 pi 2 pi 3 pi 4 pi and so forth you add or subtract that we're going to get the same for these it's going to look very similar i'm not going to talk you through it i'm just going to tell you that these are the same identities that we have here we can add and subtract full revolutions for sine cosine cosecant secant and half revolutions pi for tangent and cotangent and get the same exact value out the same output out which gives us identities that we can use so one thing this allows us to do is instead of like counting full revolutions and things like that we can really easily translate things that are outside of our normal interval from 0 to 2 pi into something on there so because we know all our periods we know that sine cosine secant and cosecant repeat all of our outputs repeat every 2 pi tangent cotangent every pi every half revolution and every full revolution counter respectively there that we can just start adding and subtracting those things and so we just get we're going to identify what trig function we have what the period is and then add or subtract mostly subtract here um 2 pi for all of sine cosine cosecant secant and pi tangent and cotangent or multiple zero to get ourselves within that that normal interval that normal domain for us and so let's try it let's try sine so we know that sine has this this period of 2 pi that means i can add or subtract multiples of 2 pi that's what that case says says any number of 2 pi's that you want 2 pi 4 pi 6 pi whatever until you get down into that interval of 0 2 pi so let's take a look at it 19 pi over 2 is that's a lot and i have to count a lot of revolutions if i wanted to figure that out but if i start taking 19 over 2 and just starting or 19 pi over 2 and start subtracting 2 pi you can really do it this way if you understand that we're in terms of pi here just take 19 over 2 that's 19 pi over 2 and subtract 2 or 2 pi and just keep on doing it so subtracting 2 that's 15 halves well what if we subtract it again that's 11 halves what if we subtract it again well that's that's seven halves well if you subtract two again that's three halves so what that means is sine of 19 pi over two is the same thing as sine of three pi over two so we've just got to figure out what sine of 3 pi over 2 is so we're going to go 3 pi over 2 that's that's on that's within this 0 to 2 pi interval that is going to be negative 1. and we're done one thing i'll do i'll let you in a little hint here um one thing i do is i look at my denominator it's like two or two or six or four or four and i look at what my period is so for sine it's two and cosine is 2 pi and tangent's 1 5. and i'll take and i'll multiply my denominator by whatever that is 2 2 or 1 and i'll start subtracting that number until i get down to something that's in my interval so for instance 19 over 2 okay so 19 over 2. i'm going to take that 2 my denominator multiply it by 2 because my period is 2 pi that's 4. that's just our subtracting force so 19 minus 4 well that's uh that's 15 that's still not within my zero two pi if i'm starting to divide by two i would just start subtracting that until i get down to my three and then three divided by two is less than two so less than that full revolution if that helps you great um if not then just start subtracting 2 pi so that's 2 minus those fractions until you get down to something that's within your unit circle we do the same thing with with 9 pi over 2 so 9 pi over 2 we're still dealing with with cosecant that has a period of two pi so i'm gonna start subtracting two pi so i know that nine halves minus two well that's four halves so nine halves minus four halves is five halves minus four halves again is one half this is the same thing as cosecant of pi over two so what's that what's that mean what's nine pi over two mean that means we're going pi over two nine times well every 4 pi over 2's is going to be 2 pi think about that 4 pi over 2 is 2 pi so if we just start subtracting 4 pi over 2 to match our denominator we're going to subtract 4 from that 9 that's 5 pi over 2. well that's still outside of this interval here so subtract again that's one pi over two and that's exactly how we're getting this you can show your work if you want to start subtracting two like we did or two pi like we showed but that's ultimately we're gonna get so cosecant of pi over two here's pi over two cosecant is one over y so we're going to get one over one just one how about tangent so tangent of 19 pi over six tangent's period is every pi and so i'm going to start subtracting honestly just my denominator until i get something within my interval so tangent of 19 pi over 6 i'm going to start subtracting one pi intervals of one pi pro multiples of one pi actually i should say so 19 pi over six let's see 19 pi over six minus six pi over six that's just pi but i'm matching my denominator so notice i'm subtracting 1 there that would be 13 pi over 6. now 13 pi over 6 is still more than one full revolution so i'm going to subtract that again that's 7 pi over 6 now that's something that's on our unit circle but i'm going to do one more step so 7 pi over 6 is right here tangent tangent would be negative one half that's y divided by x that's negative square root of three over three so negative one half divided by negative square root of three over three i'm gonna do it in my head right here you should work it out negative divided by negative is a positive one half divided by square root of three over three those twos are going to cancel you're going to get 1 over the square root of 3 or square root 3 over 3. now i said i was going to do it slightly different well what if we did one more remember that tangent it has a period every pi so we could make it so that we find some angle between here and here every time so if we subtract one more pi just one more time it puts us back right there so if i do that so that's one pi over six notice notice something make it make sense that this period is actually pi here because tangent of one pi over six that's one pi over six should be exactly the same here's pi over six one half divided by three pi square root of three over two is the same thing as negative one half divided by divided by negative square root of three over two you're still going to get one half over square root of three divided by square root of three over two you're going to get those twos cancel one over square root of three is square root of three over three it's exactly the same thing that shows you that the period of pi over period of tangent is this pie repeats every pie because of those signs that we talked about earlier you should try the next couple on your own i'm going to go really quickly through them i'm going to show you sort of my shortcut on how to do this so what i what i always do i take a look at my denominator i look at my period and i multiply by that number in front of my pi 2 2 1 2 2 1. so i'm going to take my sign sign has a period of 2 pi i'm looking at my denominator of 4 that means that i'm going to be subtracting 2 pi from this that would be 2 pi over 1 same thing as 8 pi over 4. so notice i'm taking this multiplying by 2 that's going to give me my multiples of my fractions in the same denominator so i'm going to take my 9 pi over 4 and subtract 8 pi over 4 from it why 8 pi over 4 because that's still two pi so that's why what i was talking about take your denominator multiply by two that's the fraction that you want to start subtracting so nine pi over four minus eight pi over four is pi over four that looks good that's some value that's on our unit circle as far as our angle is concerned that's going to be square root 2 over 2. we know that y is our sine of whatever angle is last one cotangent cotangent has a period of pi so i'm going to be taking 17 pi over 4 and subtracting let's see that's one one times my denominator 4 that's 4 pi over 4 so i take my denominator multiply it by the number that's in front of my period and that's the fraction you start subtracting so 17 pi over 4 minus 4 pi over 4 remember that that right there is still 1 pi that's how we're getting it well that's going to give us 13 pi over 4. remember that we can always get tangent and cotangent down between zero and pi well what that means is that you should have some number on your numerator less than four for sine and cosine cosecant secant you should have some number in your numerator less than whatever two times your denominator is always that so if i keep subtracting that's nine pi over four that doesn't look like one of these angles here five pi over four uh let's see here's five pi over four i could do that but we could go further as well remember if the period of tangent and cotangent is zero to pi then you can always subtract or add these angles until you get somewhere on this particular interval right here for sine cosine secant and cosecant it's the full rotation and so what we're looking for is for tangent and cotangent to have this number less than that number for sine cosine secant cosecant to have this number less than twice that number that's 2 pi somewhere in there so that's going to give us this pi over four so cotangent of pi over four here's pi over 4 takes my x divided by y now it's because they're both the same you're still going to get one i hope that makes sense to you i hope you're seeing the interplay on on how we can use these identities and say hey let's just start adding or subtracting two pi or pi whatever our interval is and multiples of that um you could be a little more fancy and not have to do it several times if you thought a little a little bit harder about it i gave you the most basic thing i could think of there as far as is actually doing it what we're looking for again what we're looking for is to get something that's on our unit circle with the same output it's way easier than counting our revolutions to get back to that so i hope that i've made it make sense i'm going to come back with a couple more but at this point you really should understand that for these values of our trig functions that are more than one revolution even for tangent more than a half revolution we can always get something for tangent and cotangent here for sine cosine secant cosecant on our unit circle by adding or subtracting the appropriate number of periods okay a lot of students think these are easier they probably are because we're not dealing with fractions the same thing holds remember that like cosine and secant have the same period well they're based on the same thing really sine and cosecant have the same period and what that means for that that period of 2 pi is we can also add or subtract 360 degrees so whether you're dealing with tangent or cotangent add or subtract 180 secant cosecant adder subtract 360 and get down to the appropriate kind of uses of our unit circle really for sine and cosine you should be somewhere on this full full unit circle for tangent cotangent somewhere on the half of that unit circle so take a look at cosine 420 well that that's way more than one full rotation so let's subtract 360 degrees and say we see where we land if we subtract 360 degrees that's exactly the same we can do then we're going to get cosine of 60 degrees now cosine 60 degrees is really easy to find that's going to be one-half the same thing works with secant actually secant has a period of 2 pi so if we take 540 degrees which is more than 360. more than on our unit circle right now let's subtract 360. so secant of 180 degrees just subtracting 360 it's gonna be the same output now secant is this one over x it's one over cosine so one over x gives us negative one so we can do this with radian type measurements we can do with degrees degrees are typically a little easier because we can add and subtract 360 or 180 a lot nicer than we can these fractions of radians but that's just about it i hope i made it made sense i hope that you understand the interplay between our period and what we can do with it i probably went a little overboard and did a lot of examples just so you could see the hey you know what you're adding for 2 pi 2 times that denominator pi every single time and for tangent cotangent 1 times your denominator pi every single time i i've made that clear but hopefully you see that i'm trying to end with something that's on our circle to make it easy well i thought i'd give you a little extra here a quicker way to look at how to simplify some of these angles they go more than one revolution around a unit circle or even that don't in tangent and cotangent case so i'm going to show you kind of a special trick i want to give you a basic way and you certainly can do that by subtracting the period over and over and over for the respective trigonometric function but i'm going to show you a quick way to do it i hesitate because students use it like a trick without fully understanding it but i think it's going to benefit you so i'm going to give it to you anyway so here's the deal how we can sum how we can simplify the angle of a trig function that goes more than one period around a unit circle is by by dividing it in a special way and looking at the remainder upon division has to do a little bit with number theory uh modality of number so i want to look at 11 pi over 2 now obviously that goes around the unit the circle a lot more times than just one so here's what i can do what i do is i take my denominator i multiply it by the period for sine in this case which is 2 pi and then i look for the remainder of my numerator divided by that thing so here's what i'm going to do i'm going to take a look at 2 times 2 pi now that's 4 pi i'm going to take 11 pi divided by 4 pi now the pi's are going to cancel so i'm really looking at 11 over 4. not 11 over 2 11 over 4. or 11 divided by 4. so 11 divided by 4 4 goes into 11 two times and if i subtract them i'm going to get 3. this is irrelevant this is how many times you complete the period on the unit circle that's irrelevant what matters is imagine this imagine going around the inner circle one two times where you end after that is the important part that's what that number tells you so the remainder tells you how many or how far you go past the full revolutions here this does not matter this does matter and this will tell you your new numerator of that angle so 11 pi over 2 is going to be the same thing for sine as 3 pi over 2 not over 4 2 because the period is 2 pi so that 4 gets divided again by that 2. so sine of 11 pi over two is the same thing as sine of three pi over two take your denominator multiply your period divide your numerator by that number your pi's are going to cancel and you're looking for the remainder that becomes this new numerator and it will be somewhere on your unit circle let's look at tangent so tangent of 6 pi now let's look at the denominator of that angle that's one the period for tangent and cotangent is just pi so i'm going to take 1 times pi well that's just pi 6 pi divided by pi is just 6. so i'm going to take 6 divided by 1 times pi is 1 so i'm taking 6 divided by 6. if i take 6 and divide it by 1 it goes in 6 times and there's a remainder of zero now what's this mean six pi says what this means is you'll complete the period six full times one two three four five six for pi or sorry for tangent so that period is pi and you complete it six times the remainder says where you go after that so you end at zero so wait a minute that right here is your new numerator zero times pi is zero tangent of six pi is going to be exactly the same as tangent of zero why because if you go six pi's one pi is half a circle two three 4 5 6. you end right back where you started at 0. this just does it for you so take denominator times period for your respective trig function divide it look for your remainder that remainder is your new numerator in this case zero how about cosine cosine eight pi over three cosine has a period of two pi we need to know that so all four of these have two pi all of you all of your sine cosine secant cosecant have a period of two pi tangent cotangent one pi let's take our numerator that's 3 multiply it by your period that's 2 pi that's going to be 6 pi and what i'm going to do is i'm going to take my 8 pi over 3 divided by 6 pi my pi's are going to cancel and i get 8 and i get eight pi over six so eight divided by six six goes into eight one time there's a remainder of two i don't care about this this says you've completed one full period for cosine that's two pi where you're out are past that is two pi over three pi over three past that this is your new numerator so cosine of eight pi over three is the same thing as cosine 2 pi over 3. this says how many periods you complete this is how far past that completion you are it will be starting with the unit circle this is your new numerator for pi so in this case cosine two pi over three and you can figure that's uh is negative one half because that would be the x coordinate of that now do you use your unit circle for that i'm not really not really going that far i'm just showing you how to get something that's on your unit circle you figure out the rest how about sine of 19 pi over four sine has a period of two pi so i'm going to take my denominator multiply it by my period that's 8 pi 19 pi divided by 8 pi pies are going to cancel i'm going to look at 19 divided by 8. 8 goes into 19 two times a remainder of three the remainder is what's important this says you complete two two full circuits around that that period so two rotations and then what you're gonna do is you're gonna have a leftover of three that's your new numerator that's sine of three pie over four not eight because it's based on the fact that there are a period of two pie not just one pie so it's always your same denominator that you keep this is just giving your new numerator that's on the unit circle uh let's see that would be quadrant one two three quadrant two and then you're going to get um square root of 2 over 2. let's do tangent 19 pi over 6. so tangent 19 pi over 6 we're going to take a look at tangent understand the period for tangent is 1 pi take your denominator multiply it by your period so 6 times 1 pi is 6 pi your pi's are going to cancel we're going to look at 19 divided by 6. please recognize that you do tangent different from sine tangent has a period of one pi that's half a circle it repeats every half a circle sine is two pi that it repeats on a predictable repeatable fashion so we're going to divide 6 goes into 19 three times with a remainder of one that's your new numerator so this is equivalent to tangent of one pi over six so you complete three periods not rotations not full rotations that's not the thing here is you've completed three periods your remainder is one past that so pi over six it would be the new angle that you could determine tangent of pi over 6 is going to give you that oh let's see one half over square root of 3 over 2 so square root of 3 over 3. now cotangent has the same period as tangent does so if our period is pi for cotangent 4 times pi gives us 4 pi pi's cancel we're going to look at 17 divided by 4. can you do it in your head yeah of course you can 17 divided by 4 4 goes into 17 four times it will complete four periods the remainder is one so there's a one is your numerator pi over your original denominator and cotangent of pi over 4 is a lot easier than thinking about that cotangent pi over 4 is going to be reciprocal of tangent pi over 4 so that's going to be square root 2 over 2 divided by square root 2 over 2 that's 1 or reciprocal is also 1. so that would be 1. last one cosecant cosecant has the same period as sine they're very related and so we're going to say hey the period of cosecant is 2 pi denominator times your period that's 4 pi and divide that's going to be 9 divided by 4 so take 9 divided by 4 and look for the remainder if your remainder is 0 then you get 0 that's fine but here our remainder is 1. 9 divided by 4 is 2 and one fourth two full periods and one fourth of the next one why do you get pi over 2 and not pi over 4 is i just said it one-fourth of the next period well because the period is 2 pi one-fourth of the next two pi is one-half so pi over two and that's exactly what we have here so nine divided by four four goes in that two times with a remainder of one we get one pi over two so cosecant of pi over two um well we think sine of pi over two that would be one so reciprocal is one so we'd get one here as well i hope that makes sense to practice that if we understand the period of our trig functions we can take denominator times period divide pi's are going to cancel and look for your remainder that's your new numerator with the same denominator because we have periods of 2 pi for sine cosine secant cosecant and 1 pi for tangent cotangent hope that makes sense that's a little bit extra but it'll make things much quicker for you hope that helps now lastly please remember that your signs are very important and so you should be able to determine what quadrant your angle is in just by looking at the sine of whatever your trig function is so for instance let's say that you have this um you have the the relationship that secant of your angle is negative sine of your angle is positive what quadrant would you would you would that angle be in well let's think about this for a second in quadrant one all of your trig functions are positive every single one of them in quadrant two only sine is positive and that means cosecant is positive y in quadrant two your x values are negative and your y values are positive if your x values are negative that means cosine and secant are both negative now we see that cosine's negative but secant would also be negative it's 1 over cosine tangent's also negative y and positive divided by negative is negative and cotangent's negative y a negative divided by a positive is negative so only sine and cosecant are positive here and you can work your way around and kind of see that only tangent and cotangent are positive here only cosine and secant are positive here because the interplay between these signs of our axes there positive x and negative y is going to create for you this hey negative y for these particular angles is going to say that sine and cosecant tangent and cotangent are all negative in that quadrant so going back to my question what if you knew that secant was negative and sine was positive well sine being positive says you're either here or you're here secant being negative says it can't be over here everything's positive here that would put you in quadrant two it's useful think about things like that let's think about what if cosine is negative and tangent is positive if cosine is negative and tangent positive is positive what quadrant would you lie in cosine is negative cosine being negative would be any of these two but sorry uh cosine yeah cosine negative but tangent being positive would say if cosine's negative you're somewhere over here why well cosine's positive here cosine's positive here so these are the only two quadrants or cosines negative but tangent being positive would say i have to be in quadrant three that's the only place where let's see x's are negative cosine's negative but tangent is positive negative divided by negative would give you positive you can think about things like that on what quadrant think about this on your own would uh let's see secant be positive but cosecant be negative secant positive and cosecant negative big positive video think about that secant positive would be either quadrant one or quadrant four because secant relates to cosine cosine's positive here secant is positive here cosine's positive here secant is positive here everywhere else that's negative secant positive would be in these two quadrants cosecant negative well let's see cosecant negative would be cosecant is positive here because sine is positive here cosecant's positive here because sine is positive here cosecant is negative on these two quadrants so if cosine is positive that'd be oh sorry if secant is positive that means one of these two but if cosecant's negative that's in one of these two this is only one that has both secant positive and cosecant negative so i hope that makes sense uh you really should be getting these signs down sine and cosecant have the same signs in the same quadrants so do secant and cosine so tangent and cotangent so take a look at your unit circle and just really get used to that hopefully that makes sense we're going to come back with some more identities talk about the reciprocal identities in the next video have a great day

Channel: Professor Leonard

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Length: 70min 53sec (4253 seconds)

Published: Mon Apr 26 2021