[Music] all right here we go let's take a look at the next section we're going to kind of mix it up here a little bit and start talking about polar coordinates and polar graphs pretty awesome when I think of polar obviously think of polar bears that guy's adorable waving hello at you let's do this man so we're so used to everything we've graphing on the X and Y plane but now we're going to look at the r and Theta polar plane so what is that well we start with our 0 degrees or zero radians this is our polar axis and then we're referencing this is called the Pole right here it's kind of like our origin or really it's the reference point uh it's the center of everything that we're referring to so that's our pole and what are we going to do well we're just going to draw some kind of angle off of this so we have our Theta and then we're going to pick one of these lengths along the way it's kind of like the radius of the circle so what is r r is going to stand for the length you go out in AA is going to be the direction so maybe you've done some of this before in science this is big in vectors we're going to do vectors uh coming up here soon in unit 4 but we're just talking about another way to graph points so it's pretty cool this first section I think is pretty chill we're just kind of getting used to it uh and seeing how it goes here so here's an example what is this this is R comma Theta this is 2 pi over6 so here's a polar um plane here and I'm just going to go out each rung here is like the radius so I'm going to find my pi over 6 there it is I'm going to go out to put a dot boom easy peasy That's it man uh excellent so what about a negative so when I talk about this two that means I'm kind of walking towards the angle so I'm heading towards pi over 6 so what do you think negative means well you're going to walk away from the angle so the angle here is 7 pi over 6 so let's find 7 pi over six here it is instead of walking towards it we're actually going to walk away for it so it does get kind of weird here one two three I'm actually back here so what you may be thinking is hey why would I name it negatively yeah you don't have to you can name these a bunch of different ways if you don't like the negatives you can try to avoid them just like I try to avoid Mr Sullivan um and you can do it however you want awesome so along those lines here let's name this different way so I'm going to force you to do it four different ways let's start with the positive way so I'm looking at uh 5 pi over three that's a terrible looking five let me try that again 5 pi over three and I want to walk towards it if that's my angle so what is my length well I'm walking towards it 1 2 3 4 so that would probably be how most people name it using both positive so I'm cool with that let's keep going with um uh renaming 5 pi over 3 another way so I went around this way for 5 pi over 3 that's crazy let's do it the negative Direction so could I do it negatively so sure let's keep this I still want to walk towards that same angle of 5 Pi over3 but another way to call that would be be what so another way to call this would just be PK over 3 I just went pi over 3 in the negative Direction so that's totally cool if you want to do it that way how about another way let's start to think about doing it with some more negatives here so if I'm going to walk away from an angle what angle am I going to walk away from well this is the the angle of interest and see how it comes on through here to the other side to 2 pi over 3 so I can say hey let me walk away from 2 pi over 3 and if I walk away from that so I look that angle walking away from it is back over here awesome is there another way to name it instead of calling it 2 pi over 3 sure I can double negative love it double negative so I'm going to walk away from not 2 pi over3 can we name it in the negative Direction sure I went past Pi how far past I did another pi over 3 past Pi so it looks like I'm doing -4 piun over3 so there the I got double posit positive one negative one negative double negative so there's four different ways all talking about that dot right here so it gives us some flexibility uh to name these points That's it man we just named polar coordinates all right what if I want to convert these bad boys though so what if I'm in poar and I want to go back to rectangular so I got my little r Theta here but I'd really like to get back to X and Y well check it out we can totally do that check this out so I've got this polar uh coordinate system up here so there's my polar graph can I open overlay a rectangular yes you can watch this exciting boom there it is so a point uh can be named either way rectangular or polar I love that when they overlap kind of looks like a into the Multiverse Spider-Man or something pretty intense but yes we can overlap them awesome all right here we go so let me draw like a z degree angle or zero radian and pi over 3 is something like this and let's just say six is about here on it so that's the Polar well if I want to convert that to rectangular can we overlap like an XY plane on it sure there's like the y- axis there's like the x- axis see the overlap there and so I want to know hey how far over X did I go and how far up y did I go uh oh well look at this this is one of our right triangles isn't it so we've got Theta chilling in here we've got the r here we've got all the pieces we need so really you've actually done this before earlier in the chapter where I said hey man how far is this point right here well maybe remember X is like the cosine so we did o o r cosine Theta would take us out to that direction and then how did I do the Y well that was the sign so we did R sin Theta so those are the formulas jot those down have those for conversion so can we convert this bad boy here sure no problem what's the radius radius is six what's the angle we've got pi over 3 boom done and done picking nice friendly angles from our unit circle let's do why Y is going to be 6 sin Theta so and again Theta is pi over 3 and let's go ahead and evaluate these bad boys here so what is the cosine of pi over 3 well I know that's 1/2 it's in the first quadrant everybody's cool and positive oh my gosh let's try that again2 and then what is the S of pi over 3 it's going to be the radical 3 over2 so let's clean this up a little bit what is half of six while get three what is the two here is going to cancel um just like the two cancel this to give you three he's going to cancel that give you three so we were looking at 3 radical 3 so we just found X and Y so I convert that point to X is three Y is 3 radical 3 fantastic there we go awesome I got another example here um try let's do this one again here together here so it's just following the formula so it's not bad the only difference here is got to be careful with our sign so to find the x is going to be our cosine Theta so I've got that 5 pi over4 and then to find Y is going to be 4 sin Theta so we're really going to use this formula a lot coming up here next section and whatnot uh as far as converting from Polar to rectangular oh that's supposed to be a pi over four so now just I want to make sure uh I'm cool here because now when I'm talking about cosine of 5 pi over the 4 where am I well remember you're down there in that fourth quadrant so this is actually going to be a negative so this would be negative radical 2 over2 and then when I'm looking at sign sign again is also negative over there so it's negative radical 2 over2 and then when I clean this up a little bit um the twos are going to cancel so I've get -2 radical 2 this will also be the same thing it's going to cancel two of those -2 radical 2 and let me write that out nice and neat because that looks really sloppy so we've got 2 radical 2 -2 radical 2 and I always double check does that seem like the right quadrant yeah I went over and down so I'm down here somewhere sure that's that angle going through it so I just kind of make myself R rough sketch to check it out rock and roll so if we can convert one way can we convert back the other way well I hope so that'd be bummer if we couldn't so now I'm giving you X and Y so remember this is X and Y so it's kind of weird uh switching back and forth here X and Y this one I definitely want you to draw like a little sketch I I think it'll really help so just to get a ballpark I'm over 30 five I just really want to know what quadrant I'm in that kind of cleans things up so can I bring this back well let's come over here to formula land and let's say we were going to go uh let's do a point in the first quadrant if that's cool so if I had a point out here and it's X comma y so I went over X and up y again so I went up y well guess what we're back to that triangle aren't we spoiler alert even though if I did it XY I'm still got the angle created here I've still got R well what relates these bad boys together well I can do Pythagorean theorem A2 + b^2 = c^2 so X2 + y^2 is R 2 but I'm looking for R so what do I got to do let's just square root that so we're going to square root that so really come over to your formula this is going to be the Square t of X2 + y^2 awesome and then down here in Theta how do I find Theta well that's just an inverse tangent isn't it so I want to do inverse tangent opposite over adjacent so I'm going to take the inverse tangent of Y overx there it is there's your formulas so now we got them let's use them here so I want to find r r is just going to be -3 2ar + 5^ 2 and then I'm going to square root it there it is right there what is tangent tangent is going to be inverse t of YX of 5 -3 awesome so I got them all set up now let's just clean it up and have our answer here so uh can we Square this -3 s is 9 5 squ is 25 and I'm going to square root it so I'm really talking about what here the square root of 34 so kind of a weird number don't stress out we're get some weird radicals here um how about the next one I think we're going to have to go calculator all right so here we go let's take the not tangent let's take the inverse tangent of 5 / -3 and we're going to get this answer right here negative 1.03 so let's shot that bad boy down here so we're looking at let me get my pencil 1.03 and I don't know have to write that last zero but I'm such a trunk hater I like to go three decimals and just stop writing so I'm going to leave it in there um awesome so this looking pretty good except did you notice this well I have to be in between 0 and 2 pi so remember the calculator is only going to give you inverse tangent between one and four quadrant that's his restricted domain so I'm not in there with that negative so what do I got to do well it's weird let's just take a look at what was negative one would be something like this that's not even where the point is that's why I drew this the point's up here so if the point's up there how far away am I we really need to add pi to this so we need to add pi to this right here so that's the trickiest part here is making sure that the calculator G gives you the point of Interest so let's add pi to this bad boy and see what we get so we're really looking at what 2111 I like that number I can remember that so what is my final answer after all this so the R value is radical 34 and then my radians or my angle is 2.11 so again that little picture saves me with this tangent that to me is the hardest part is just remembering if I got to do something there awesome let's try another example so again just before I get too far I like to just say what quadrant am I dealing with here so I'm over two down six I'm in quadrant four so you just got to make sure when I'm done everything matches up let's just do it's a massive plug and chug so once you have the formulas it's really just go for it man so we're looking at the square root of 2^2 + -62 awesome and then let's clean it up I'm think I'm talking about 4 plus 36 here when I Square them and that is radical 40 does this break down a little bit Yeah I think four goes in there so I am going to clean this up a little bit I can say this four goes in there 10 times so Square OT of four is two so I'm going to say two radical 10 so I will clean up my radicals um if I can make them look a little nicer there awesome now the tough one here the tangent so we're trying to find Theta the inverse tan is not the hard part that's just plug and chug man so we're going to put y overx so we're talking about -6 over2 the hard part is am I going to get the right answer answer so I can look at this am I going to get the right answer well yes it's in the right quadrant tangent is going to hook me up so that's looking good here so let's do it it's going to give me the angle let's clear that bad boy out so I believe I have inverse tangent of -6 ided two except is that between zero and 2 pi no so what do I got to do to this I got to add the two Pi so I got to make it fit so I'm going to add let's add 2 pi here 2 pi that just brings me back around the circle 5.03 4 so in this case we had to say oh man yes it gave me the right answer but it wasn't uh it was in wasn't between zero and 2 pi so I actually had to add two pi to get back to it so I don't want the negative version of this angle I don't want this I want to go the other way so that's why we had the two Pi so in this case Theta was going to be 5 point I forgot I thought I could remember oh I would have guessed that 5.03 4 all right 5.03 4 so if it does re tell you where theta's got to be you got to put Theta there so what is our final answer here we've got two radical 10 is our R and then our radians our angle is 5 let me clean got oh oh whoa whoa whoa 2 radical 10 and then 5.03 4 awesome and that is our final answer there there it is right there that's our point just converting things over sweet so we're about done here that's half of it but we do have another way to graph points and we've done this before along the way either Algebra 2 or INR math 3 complex numbers remember a plus b i so if you remember this this was a real a was the real Part B was the imaginary part so together they are complex there are two parts they are imaginary and real so we say they're complex number so we had these formulas here where you could plot a point A Plus bi I you could find the absolute value of it which really was what let's take a look at that on the picture if I plot this point it's just a real and imaginary plane so this when we're Square rooting negatives we kind of get these imaginaries so you go over three four they're great because they're just like rectangular points except it's the x is real and the Y is imaginary so sometimes even we call them X+ Yi you may see that as well too because we're doing the same thing as a plus b I and then the absolute value what is the absolute value well it always goes back to this right triangle here so if I draw this in what is a and b well it's just X and Y uh and you're finding what this distance here which you're really finding R so the absolute value here um is just that distance which is just finding R so it's all the same stuff here awesome what if I ask you to name this point yeah no problem you went over 1 2 3 4 so we got 4 and then you went down 3 I rock and roll so that is a complex number on the complex plane so what do you think we're going to do to it we are going to go complex polar woo I love it we got to we got to put it all in there so if I want to go to complex polar I got to get R in Theta so it's the same actual formulas here that we were doing before if you like a you can use a b but you're find the absolute value uh you're going to do the same thing it's just going to be I'm going to keep it with X and Y if that's Coolio x^2 y^2 and then what am I going to do here I'm going to inverse tangent so it's all the same stuff but we're just going to write it out a little bit differently um when we come to The Final Answer here so this is complex has I it is in rectangular I want to go to Polar so let's do it let's find that R is going to be the square root of -32 Plus radical 7 ah radical 7 squared which is nice I mean I I kind of picked that on purpose because that's nine well happens when you square a square root you just get back to where you were so check this out I thought let's make one workout friendly here so 9 and seven is 16 16's cool because I know the square root of it so it's just going to be plain old four like that awesome awesome that six got worse and worse as we went there I can't leave that I can't leave that there okay now let's go to Theta what is Theta here well we're going to go inverse tangent and we're going to say y overx so we're going to say radical 7 over -3 again I kind of like to know well where am I it's a question I ask myself a lot where am I well I am in what quadrant I went left and up I don't even care how much I just got to know that I'm in quadrant two so is this going to give me the right tangent no tangent is always one or four so if it's over there I imagine it's going to give it to me over here so you're going to have to do what what to it you're going to have to add pi to it so just be careful it's not going to hook you up um let's see what that is here we've got uh let me go back to where my calculator go there he is there he is so we're gonna say it is clear that out all right let's type this bad boy in I'm going to go inverse tangent of radical 7qu < TK of 7 and then I divide that bad boy by -3 and let's see what the calculator tells me. 722 so again I'm a huge trun haer I'm not going to worry about rounding but I got 722 don't forget to add that pie there though because I it did it gave me down here but again I want to be up in the second quadrant so we got to be Coolio with that so we're going to add that pi to it so let's go back and say add some Pi put that in there and what do we end up with 2.48 that's an answer I can hang with uh 2418 now how we going to write this out so that's Theta right there so how am I going to write that out well it's kind of crazy we are going to say we're going to write it out as that whole R cosine Theta R sin Theta so we know R is four cosine of 2.48 we know um this is going to be for sin Theta but because it's the imaginary part it's actually 4 I and I'm going to take the sign and I'm going to run out of room 2418 there it is right there that's a monster sometimes what we do to make a little bit shorter is take the four out of it both see how they both have that four so you may see it written like this I wrote write most of my answers like this 4 cosine 2.41 8 oh my gosh I'm having a terrible time with my handwriting today plus I took the four out so there's the I so it doesn't save a ton uh but I want you to be able to recognize it back in the day I swear we used to use CIS c i for this and that cleaned it up a lot but there it is right there oh my goodness close the brackets that means just distribute the four there it is right there holy cow why don't you pause it and try this one and see how it goes good luck all right there it is right there what a monster I love it I love it hopefully uh you notice that this was in the for I made a little sketch here I hope it didn't trip you up that it's just negative one I so if it's not written it's like a negative one so don't freak out plot it's in the fourth quadrant so it gave it to me the fourth quadrant there was nothing limiting me between zero and 2 pi so I left it that way if you had a two Pi it's totally the same answer but you didn't have to it's just right there that's a big dog answer very impressive if you got that right there all right let's wrap it up the finale is just going backwards and we are done with this uh so here I am in the rectangular complex number and I want to go back to Polar so really again it's the same formula so once you have this and once you're done with the section you're going have this bad boy memorized so we can just plug and chug back into this R cosine Theta R sin Theta and we can see Theta right there there's Theta I can see R is right there so really this was probably my favorite we're going to say what is X it's just going to be what it's going to be the three ah cosine of pi over 3 so it's really the same stuff again it's just kind of a little different notation because we got the I in there and then let's change colors for the Y so for the Y I'm going to say it's uh 3 sin pi over 3 just be careful with your signs here uh make sure we're good on everything here so let's evaluate the first one oh first quadrant so nice pi over 3 is just2 down here uh sine of pi over 3 is radical 3 over 2 still not too shabby right there so what is half of three well it's just three Hales and then what is that well nothing really simplifies so it's 3 radical 3 over 2 so that gives us our final coordinates of oh not coordinates like that we could write them like that as long as you specified but I'm going to write them as a plus bi so I'm going to say it's three halves plus where does the I go it actually we don't like to put it after square roots so I always try to put in front of square roots because it we don't want to get underneath that square root long story short there's the a plus bi I something like this one more just to make sure we're cool uh pause it and try this bad boy see what you come up with all right I tried to kind of pick a weird one here that had a zero in it so hopefully you got the cosine of zero just gives you the real part is zero here and then over here you got the netive 1 so gave you ne4 part for the imaginary so this is just plain old imaginary number nothing complex about it's just4 I that's it the section had a little bit everything complex real imaginary trig I love it mix it all together uh good luck on the practice in the check peace out e e all right there it is there I tried to give you a weird one where you got a zero which is kind of nice uh hopefully you got this4 plus zi but does anybody really write that no way we just say it's ne4 so it's actually just a real number here is all that bad boy is That's it man we got complex we got polar we got rectangular we got it all together this mix a little Trigg in there what a what a section here so good luck on the master check hope it goes well peace out