continuing with module three which covers the third homework into third quiz hopefully we'll finish module three this last thing we're going to do is something pretty different and noon it's called trig graphs and as we start creating these trig graphs you're going to see that one of the things I use is the unit circle we just had a couple lessons on unit circle once again here is the document that remember the unit circle allows us to easily determine the trig functions of any of our special angles in other words this chart goes from zero degrees all the way to 360 degrees or in radians 0 to 2pi and the special angles are the like the 30 45 and 60 and then you keep moving that same distance all the way around and the whole key to why the unit circle is so helpful and powerful is that for instance if you have a 30 degree angle or a PI over six angle this point is what helps to form that angle and if we want to get the trig functions we've learned before that if I want the cosine of PI over 6 I simply look at the x value of this point and that equals the cosine of PI over 6 the cosine of PI over 6 would be square root 3 over 2 if I wanted the sine of PI over 6 or if you look for the sign the sign is always the Y value of the point so the sine of PI over 6 would be 1/2 if I were to say go to the third quadrant say I was given I was told find the trig functions for a 7 PI over for angle let me just start here you go all the way around to right here once again the key is you just have to know what the X and the y are since the X the square root of 2 over 2 I would say the cosine of 7 PI over 4 is square root 2 over 2 and I would say the sine 7 PI over 4 is negative square root 2 over 2 so now by using the circle we're going to do something different now these are called trig graphs and here's what we're going to do I'm going to first make a table and put some numbers inside this table and then once I have the table then we're going to create another graph not not a graph like the unit circle but a different kind of graph so what I want to do I'm going to start out with an angle since we normally when I do angles I call it angle theta what I'm going to do is I'm going to pick an angle and for that angle I then want to determine the cosine of that angle and then also the sine of the angle as you know by now these are really the two main trig functions cosine and sine everything else we can figure out if we know the cosine and the sine here's what I'm going to do now in a sense I'm going to start and I'm going to do this and I'm going to do this in radians I'm going to start at a zero angle and then put down so what is the cosine of zero well the cosine of 0 is 1 and the sine of zero is zero how did I get these so quickly it's because I know the unit circle angle zero is right here I know this point is 1 zero the cosine of zero is one the sine of zero zero and what I'm going to do is I'm going to start putting in different angles that one because I'm going to Vinci move around this unit circle I'm just going to put in some different angles and then write down the cosine and sine I'm not going to choose every angle matter of fact I'm going to keep it really simple you know what I'm gonna do I'm gonna choose what I call sort of the really simple major points I'm gonna start here at 0 then I'm gonna go to PI over 2 and then I'm gonna go to PI and then 3 PI over 2 and then up here to 2 pi and then maybe for fun I'll go around again and do a few more let me just so what that looks like so I'm gonna go ahead and look at the angle PI over 2 and then I want to look at the angle PI and 3 PI over 2 and then 2 pi now if I keep going there's just a couple more points for fun if I go if I'm a 2 pi here now and I want to go up here again I add PI over 2 to 2 pi which ends up being 5 PI over 2 and if I go another oops 3pi let's call the round twice so now so I went around the circle twice 0 to 2 pi then 2 pi to 4 pi now let's figure out what the cosine is sine is for all these angles and this comes directly from the one the unit circle cosine of PI over 2 PI over 2 I know it's like the top here the cosine is the x value so I know it's 0 the sine is the Y value so I know it's 1 I have the angle PI angle PI is right here I know the cosine of PI is negative 1 and the sine of pi is 0 now once again I'm showing you this unit circle but I hope you realize on the midterm exam and the final exam you have to really have this unit circle memorized so that you can know the trig functions of these special angles so if I go to 3 PI over 2 3 PI over 2 is down here so this is my point 0 -1 so the cosine is 0 the sine is minus 1 now at 2 pi I'm back over here where I started the cosine is 1 the sine is 0 if I go around again actually you're going to see that these points sort of repeat 5 PI over 2 is really the same as PI over 2 remember we called those coterminal angles they actually start and stop at the same place 3 pi is coterminal with pi so they have the same trig function 7 PI over 2 it's just like 3 PI over 2 and a 4 pi is just like 2 pi so I've created this table once again distribu I'm looking at various angles as you move around the unit circle and for each angle and figure out what's the cosine of that angle and what's the sine of that angle and for me the easiest way to determine these was just to use unit circle here's what I'm gonna do now now I want to go ahead and I want to graph now let me do this this could be a little confusing but we have to do it because this is how the book does it it just made sense to me that when I created this table my angle was theta and they here my trig functions however what we're going to do now is in the book instead of using theta they still like to use X where X is now an angle they just like the variable X so in a sense of time called the angle theta we're gonna call the angle X so this is really cosine of X sine of X and here's what we're going to do we're going to make two separate graphs and the equation we're going to first graph is why equals cosine of X or basically here's my X and this is going to be the cosine of X or I can also call this Y so you know what it's almost like let me just I know this can be confusing let me do a quick example if I were to give you an equation y equals 2x plus 1 and I said let's go graph this here's what you would do you put the X here I mean this is one way to graph it a common way to create a table and you choose some values for X if X is 0 Y is 1 if X is 1 Y is 3 if X is 2 y is 5 maybe we'll choose some negative numbers if X is negative 1 Y is negative 1 if X is negative 2 right and then we could go and graph these points right 0 1 X 0 y 1 1 3 2 5 negative 1 negative 1 negative 2 negative 3 and my graph is not perfect basically I should have a straight line well let's do the same thing now instead of the equation y equals 2 sine X plus 1 let's do the equation y equals cosine of X if I were to do that it's the same thing I'm going to choose some X's and figure out my Y's so if I choose x equals 0 Y is going to be what is the cosine of 0 0 radians well I know what's 1 the next one I'm going to choose because I know this is an angle it's in radians I'm going to choose PI over 2 what's the cosine of PI over 2 and if you look if I kept on going here this right here is basically the same as these two columns here so what I'm going to do is I'm going to graph now this is going to be my X and this is my Y so this column here are the X's this clumb here is the Y and this is for y equals now here's what is a little tricky normally like over here for this graph when it made my X my axis is 1 2 3 4 5 well look at my exes here it's these funky numbers with PI in it how do we how do we even I mean I I can draw my X and y-axis but but like am I supposed to get a calculator and figure out what pi over two is 3.14 divided by 2 well no what you do is you go ahead [Music] since these X numbers all have this pi in it here's what we do I'm going to I'm going to basically graph this thing up to 2 pi I'm not gonna do all these so I'm going to say 2 pi is here so 0 C R 2 pi C R what is half of 2 pi well half of 2 pi is PI so here's pi what is half of pi well half of Pi is PI over 2 this is PI over 2 and what would it be if you go halfway between PI and 2 pi and it turns out zero PI over 2 PI over 2 2 pi that's all my exes are here 0 PI over 2 pi 3 PI over 2 2 pi so now what I'm going to do is now my I noticed with my values of why it goes between 1 and 0 and minus 1 so while just from my y-axis have 0 1 and minus 1 and now I'm going to go plot these points those like I'm going to plot X 0 y 1 x 0 y 1 it's right here then X PI over 2 y 0 so X PI over 2 and y 0 now when X is PI Y is negative 1 so here's PI except the Y is negative 1 what happens at 3 PI over 2 y is 0 what I puts at 2 pi it's 1 let's go on we can go out a little further how about we go on to 5 PI over 2 which is 0 and maybe right here be 3 pi 3 PI is negative 1 if I'd wanted to I could have even gone negative and I could have had like negative PI over 2 negative PI over 2 negative PI over 2 remember negative angles you go clockwise so negative PI over 2 is actually right here so it would be negative 1 all right so let's try and combine whoops let's try and combine these points and see what kind of graph we have now please I haven't graphed every single point but what I'm going to show you here is the general shape you don't just connect these points that go straight lines instead these are like curves it's not always easy to let me try and draw this it's not always the easiest thing to do to make a look I'm sure the book has nice nice graphs but I'm going through every point up something this is wrong negative power to the cosine is zero so you know what happens and this goes on forever and this goes on forever and it's like oh they call it a wave my practice called a sinusoidal wave where it just goes continually up and down the same distance and it just goes on forever this is what you get when you graph the cosine function it's very important right away the easiest way to tell whether you have a cosine function is you look at the angle zero in this case at angle zero my graph has a value of one I know that only happens for the cosine as a matter of fact let's go ahead and based upon now this second column these would be the Y values if I'm trying to graph y equals sine X in other words this is a graph of y equals cosine X let's do one of y equals sine X now if I wanted to X of course is an angle if I wanted to I could do all my points when the angle 0 what's the sine of 0 the sine of 0 is 0 if I go up to PI over 2 what's the sine of PI over 2 sine of PI over 2 is 1 I could keep going around and watch the sine of PI but you know what if I were finished at this table it would look just like here's my X's and here's the Y's so once again I've really already done I've already created this table so now let's go ahead and plot this one so my x-axis is like my angle and let's go ahead and call this right here 2pi so half of that would be pi after that's PI over 2 and then here you have 3 PI over 2 when you keep going 5 PI over 2 and then 3 PI and over here I have negative PI over 2 negative PI whoops so now I'm going to now plot these are my values of X these are my Y's which is the sign of X so when x is 0 Y is 0 towards the sine of 0 is 0 I go to PI over 2 the sine of PI over 2 is 1 so I'm up here and then the sine of PI is back to 0 but then the sine of 3 PI over 2 is negative 1 and then the sine of 2 pi is back to 0-5 a few more points the sine of 5 PI over 2 it's back to 1 sine of 3 pi is 0 I look at a couple of negative angles negative PI over 2 once again negative PI over 2 min to go in the clockwise direction to right here what's the sine of going to be negative 1 and then the sine of negative PI it's back to zero so now if I try to connect these dots and once again it's going to be a curve they'll just draw on straight lines but look something like this I'm gonna really turn this around to try and make this look decent so here's the kind of graph now if you compare if we can see them so at the same time notice the cosine this is very important the x-axis is the same same point same everything we chose the cosine of course when the angle is zero cosine is up here at one where's the sign is that at zero so the cosine starts out at its high point and then it drops down and then it comes back up and then it starts going back and forth the sine starts at zero it goes to the high point and down and back so those are the two main trig graphs and once again what have we done the X it's my angle the Y is in this case that's the sine of the angle and we simply went to sort of the four major points I need a circle we got the data and we made our rough sketch of a graph so those are already the two key trig functions but what we've gone over now is the very very basic in other words we looked at y equals cosine of X where X is an angle and y equals sine X well these are the very very base and most of times this is not what you end up having their problems instead you'll have things like sometimes you have a coefficient in front a great trig function sometimes you have a number in front of the variable X inside the cosigner and there's all kinds of and you can have even a number tacked on here at the end sometimes shop negatives so the question is I know the graphs are my very basic functions but how we're thrown in these extra numbers begin to affect my graph and it's very easy to determine this it's not very hard matter of fact here's the to sort of standard equations you have a a is gonna be like a number a times cosine this is really a Greek letter it's not W I think it'll speak Omega Omega X the equation we have for the sign looks the same now what happens is we give a a special name actually the absolute value of a so it's always going to be positive maybe you've heard of or like this the amplitude the amplitude means how far is it if we think of like the middle of the graph in terms of up and down the middle of the graph in this case is my x-axis the amplitude says how far is it from the middle of the graph to the highest point so that's what amplitude is if you look at a graph the amplitude is just how far from the middle of the graph to the highest point or how far from the middle of graph to the lowest point it should be the same number and it's always positive when you give an amplitude you always just say it's a positive number so that's what the a is this Omega in here or it looks like a little W what it does we can use it to do a calculation for this large T which I'm going to explain in a minute because T it's called the let me go look at the graph explain the period the period is how far you have to go in the x-direction until the graph begins to repeat this these kinds of graphs just keep on repeating forever and ever now but the question is how far do you have to go before it starts repeating usually the easy thing to do is to check like here it's pretty easy I could start at the top here then it goes down and comes back up and you come back to the top and you know what that tells you what your period is I would say in this case if I go from PI over 2 to 5 PI over 2 I've actually gone a distance of 2 pi that would be my period the period is how far it takes for the graph to begin to repeat itself so if you look at a graph once you determine the period you can plug the period in for T and this gives us a way to calculate Omega I could say if we move things around the Omega or that W equals to PI over the period so for instance for this graph since appear it's 2 pi 2 pi over 2 pi omegas 1 and Omega is the number that you multiply the X by that's in a cosigner sign and since it's 1 it just looks like this so let's review that here are my two possible trig there are actually more trig graphs in these but we're just going to focus on cosine and sine this is like the general equation y equals a times cosine Omega X y equals a times sine Omega X first step is to figure out whether your graph is a sine or a cosine and once you figure that out you can always find the a the a is the amplitude from the middle of the graph how far up or how far down does it go that's the amplitude the period is how long it take how low it takes I should put that there for period how long it takes for the graph to start repeating and when you figure out that period that gives you a way to calculate this Omega and then since you know a and Omega then you're able to actually determine the equation so basically we have two things we have equations and graphs so now we're gonna take some time we're going to first look at some equations and based upon the equation we're going to figure out the graph if it after that we're going to look at some graphs and from the graph we're going to try and write the equation for that graph alright since to a couple examples real quick first let's just take a couple of equations and try and create the graph so let's say why equals two cosine and about we're asked to create the graph of this here's what you do of course you need to know the amplitude amplitude is easy right it's always just a number and for the trig function and I don't care if I numbers positive or negative you just always make it positive amplitude is always easy and remember here's my standard equation let me write here since a is 2 and Omega is also 2 since a is 2 absolute value of a is the amplitude observe a 2 is just 2 a patoot remember we called tidak period and here's this little equation you have to memorize that for the test would that be told that so the period equals 2 PI over Omega from our equation Omega to 2 PI over 2 so now we've determined that the period is pi the amplitude is 2 now I'm going to since I'm when I get ready to graph I'm going to take this period and do one more step I always take the period and I divide by 4 because when I divide that period by four that sort of tells me where I should put my steps or my marks for my graph in other words you know when I did these graphs I went from zero to two pi well here the period is going to be how far I'm gonna make my graph now in a divided by four this tells me what are the sizes of the pieces there were it's PI over four once I have a period I always break it up into four equal parts I break it in half and then in half again this number here PI over four it's going to be the step size or how much you move that word this is zero this is PI over four if we go PI over four more weight over PI over 2 and this is 3 PI over 4 the reason I break it up into four equal parts is because as i graph it now these are the points where I'm either at the height or the bottom or maybe passing through now since this is a cosine first of all now okay also second thing in dimension just my amplitude is 2 now in my graph I start of putting a 1 there and I draw my graph that's going to go all the way up to 2 and to negative 2 so since it's a cosine remember when I first did the cosine graph my first graph cosine always start I it goes low then comes back high right cosine is 0 if I put in 0 in for X cosine is 0 I don't think it's just two but I don't need to even look at this equation now I just look at these points here and I realize a cosine starts high at the next point it hits zero and if you keep going it goes down to the low point then if you go it comes back to zero then if you keep going it goes back up hi if I were to connect these in fact one two I could keep this going on and it keeps repeating but this is adequate this is the graph of y equals 2 cosine of 2x so let's review that again I'm given this equation I'm given a cosine equation cosine function right up here I write the general function I realize a is 2 Omega is 2 once I know a I know my amplitude so I know how high and how low my graph is going to go that's really always very easy then the period I have this little formula to use for the period 2 PI over Omega since I know Omega is 2 here it ends up being PI and when I get ready to graph it I know I have to break up this PI into four separate 4 equal parts so I divided by 4 so I know as i graph this thing you're gonna move PI over 4 so I did my PI here I cut this up into four pieces PI over 2 PI over 4 3 PI over 4 and then because you know it's a cosine cosine an angle of zero starts out high in this case that the next point it goes to 0 the next point it goes low then back through zero and up to the top let's graph a let's graph a sign about so we're asked to plot or graph make a rough sketch of y equals 4 sine of 1/2 X the first thing I sort of say to myself is I have to remember sort of what the sine graph looks like remember the sine graph when the angle 0 starts at 0 Gosai comes back goes low back high and keeps on going but the key is once again as for omegas 1/2 we were to try and graph this I did you know the amplitude of course an amplitude is the absolute value of a and pitch is going to be for the period t is 2 PI over Omega Omega is actually 1/2 if you simplify this it's a being poor PI now remember I'm gonna do my remember I get my period divided by 4 so every step is going to be like pi so Sonoma amplitude is 4 I'll just make this mark 4 and this negative 4 course this is zero if I know my period is PI I just go maybe right after here of course I know if I cut it in half PI over 2 if I cut that in half PI over 4 3 PI over 4 now since I know it is a sine wave I know it starts at 0 and my first point it goes high comes back to 0 then goes to my low point and back to 0 so if I try and draw this now and there's your sign wave amplitude of for a period of Pi right alright so I hope you will not find those too hard now in the it's a couple little twists I think they throwing the homeworks let me just discuss the impact one of the things they will do is three cosine of let's just say so the difference is now up until now we've had equations like this now at the very end we add on a number this number ends up pretty straightforward it ends up it causes a vertical shift here's my advice about doing these graphs the vertical shift I'm going to include at the very end so it's almost like for now I ignore this I'm just going to graph this but then I have to remember at the end to include my shift and I'm gonna have to sort of redraw it but for now let's just figure this out so a is 3 which is my amplitude Omegas 4 so the period is 2 PI over 4 which is PI over 2 and if I divide by 4 so actually Palmer 8 it's gonna be my steps for my graph so let's go ahead and draw my graph let's get into cosine so my amplitude is 3 so I'll just mark 3 up here negative 3 my period it's going to be PI over 2 there's PI over 2 and of course when I draw my period I break it up into 4 equal places PI over 8 so I start here PI over 8 and then this is 2 PI over 8 which is actually PI over 4 that's 3 PI over 8 and now because it is a cosine wave I know cosine start I then it goes to 0 it goes low then back to zero and it in time but what I'm gonna do is normally at this point I draw in my graph but because the very last step is I have to account this vertical shift all I've got so far is this function right here so I'm gonna plot that but I'm gonna do it like in dashed lines which tells me it's not really the final answer it's just like a temporary intermediate graph in other words if I can do this what this sort of lightly so there's the graph of simply three cosine of 4x the vertical shift is very simple if the vertical shift is like this a positive 2 that means you take the whole graph and just move it up to the whole thing if this would happen to have been a negative 2 then you take the whole graph and move it down by 2 so in other words what's going to happen is 3 is right here this is going to go up to like maybe like 5 it's zero here it's gonna go up to right here so I'm going to draw my new points just negative 3 here it's got to go up to but right here this one's going to go up to here and this enclose up to here and now I go ahead and draw in my sort of new graph so then this right here is my final all right so the new thing here was a spherical shift pretty straightforward you sort of ignore it at first figure out your graph and you come in at the end and do this shift now one more thing then we're all done with these trig up we're actually not done well one more thing you got to see it might be something like this let's have a couple new things how about if I do what if we say to graph this well there's no vertical shift but my a right is my a is negative one the Omegas 2pi that means the amplitude amplitudes the absolute value is going to be one the period which is 2 PI over Omega 2 pi over 2 pi the period is 1 and what's interesting is up until now if you look like all the examples all the periods we calculated all had a PI in them and because of that here this period was PI this period was 4 pi but sometimes they make it like this where because they stick a PI into the original equation end up with the period just being a number I'm still going to divide by 4 so you get 1/4 and then you go here first of all the amplitude okay now there's no what's interesting the amplitude is one but this - up here is going to change things and what I'm going to do once again is I'm going to first ignore the minus sign and I'm going to graph sine of two pi X because what the minus sign does it causes what we call a vertical flip around the x axis let me get that in a minute let's first I'll do a dotted line graph of sine of two pi X so first of all amplitude it's just one period is one so if I break it up into my four so to have 1/4 1/2 3/4 and 1 now since it is a sine wave sine wave starts at 0 goes high back to 0 goes low back to 0 so this right here is the graph of sine of two pi X now to include the negative what's it mean you'd a vertical flip around the x axis I mean you take this graph and you sort of rotate it like hundred eighty degrees you flip it around the x axis so is this whole thing if you can imagine this thing flipping over then it's going to end up being like this right if you look at the dotted graph if you just flip it and then this right here is finally I graphed for that equation now the last thing I want to do and this woke LIGO fairly quickly is also on the homework on the quiz sometimes everything I've done so far is taking equations of these trig functions and graphing them and by the way I have focused solely on sine and cosine if you go to your book they give you examples and walk through how to graph tangent functions and even secant and cosecant cotangent I don't have the time to go through all those and I don't think there's much of those on the homework but it is in the book if you're interested or need it for something here's Harmandir now now let's take a graph and figure out the equation most the time the first step you do is you have to determine is this a cosine wave or a sine wave and the whole key is what happens when the angles 0 so you look at here here's my axis 0 PI 2 pi 3 PI 4 PI up and down 0 to 4 here's my starting point when this thing starts my graph is high well what is that cosine a sine this is where you have to remember you go way back to our first graph remember here was the cosine graph how does the cosine start out starts out high this must be a cosine graph what I'm going to do is now I know the equation for the cosine is this there's no vertical shift really no what I need to determine if I can figure out what a is and what this Omega is and I've got the equation I just have to figure out what a is and omegas find numbers for those amplitude equals a amplitude is really always easy to determine from a graph because you once again it's the distance from the middle of the graph to the highest point or the lowest point hopefully you can see this goes up to four and down to negative four hopefully it's pretty clear that the amplitude is four which means a is four so I know a now if I can figure out oh make ax then I'll have the equation how do we find Omega well we have this equation for the period which is two PI over Omega if I solve for Omega I can really say Omega is two PI over T where T is the period so if I can plug in my period I can calculate Omega well I can look at the graph and find the period remember I said the period is how far it takes for the graph to begin to repeat easiest thing to do is you start here at zero and it's going down so I'm going to keep tracing my graph until I get to the top and go down again so here go down down down I'm at the bottom I come up I haven't started repeating yet I come up I come to the maximum and then start down again at this point now I'm beginning to repeat what's over here okay so that means the period the distance where it starts to repeat is from here to here which it looks like on this graph it says 4pi so T is 4pi so now if I plug that in here Omega would be 2 pi over 4 pi which is 1/2 so now I've got my a and my Omega I can say the equation for this graph is 4 cosine of 1/2 X