Ch 8 1 Polar Coordinates

hi class we are starting chapter eight chapter eight is all about polar coordinates we're going to start with 8.1 which is our primary discussion on polar coordinates so what polar coordinates are is basically new way to look at the same points that we've been working with already in what's called our rectangular coordinate system okay so the rectangular coordinate system also known as the cartesian coordinate system is the standard way that we plot our points and our functions on graph we've been doing this the entire course so far we move over and we move up right that's rectangular coordinate system or also known as the cartesian coordinate system there are lot of other ways of plotting the same points okay there's what we're going to study right now which is polar coordinates we have what are called cylindrical coordinates but these are typically reserved for three dimensions spherical coordinates which are also reserved for three dimensions parametric equations which we're going to look at later on in this chapter as well and vector equations are another way of representing points in space okay so in this chapter we're going to focus on polar coordinates right so instead of using and to get point on the graph we're going to use radius and an angle right so let's say have my regular rectangular coordinate system if want to plot point on this graph know that stood at the origin and move left or right units okay so this is the point that want to get to start here and kind of think of these as city blocks right like can't go up the middle so kind of go along the edges to get to my point so move over left and move over right so i'm going to move over some amount to the value of this point right and then i'm going to move upwards units until get to that point and that's my point that's what we've been doing so far right i'm sure you're very bored while did that because we've done this so much already but now if want to get to the same point in terms of radius and an angle still start at the origin but move over whatever the radius would be to get to this point okay that would be my radius and then what i'm going to do is i'm going to sweep an angle until get to my point right so move over distance of and then sweep up angle of theta and still have that same length right so still have and sweep up until get to the point that want and move an angle of theta to get there so i'm looking at radius and an angle in order to get to this point instead of saying i'm going to go over and then up right go out whatever the distance is from my center to that point that's my radius right go out that amount and then sweep an angle upward now plot my points instead of and it's and theta okay so again need radius and need an angle and that's the order that they come in radius first angle second and if you notice that's kind of what need in order to get to this point right first need to know the distance from the point to the origin the radius and then once move out that distance then swept my angle so first thing need is the radius and then can figure out where and then need the angle okay so and theta so that order radius first theta second we're used to doing everything in and so obviously there's some sort of connection between and theta and and right so since we're used to doing everything in and it's nice to have relationship so that we can talk about how to go from our coordinates into and theta coordinates okay so if want to go from rectangular to polar right so i'm already in and want to talk about that in polar coordinates we need to understand that the radius is the length between the point and the origin right so the distance is squared plus squared and it's very easy to see that as well because you can see that it's really just the pythagorean theorem we saw this lot in trigonometry where well one where we need to find the hypotenuse of this right triangle okay but also the distance from here to here is going to be given by minus zero and minus zero both of those squared added together square root right so it's always just squared plus squared square root the angle is found by tan inverse of over and again this is using the inverse trig forms right because we have and given to us we had to calculate the radius so we generally want to use the values that are already given rather than values we have to calculate when trying to do any kind of math right so instead of doing sine inverse or cosine inverse we're given and so we'll use those values to find angle by using tan inverse okay so radius is found by the distance between the point and the origin and the angle is found by tan inverse of the height and length of the triangle if we want to go the other way from polar coordinates to rectangular coordinates the relationship is that is cosine theta and is sine theta and that should make sense as well because again we have our triangle right this is why it triggers so important because it's everywhere so this radius is length of it's not unit value right so if want to find this value remember is always cosine of this angle but i'm also stretched by value of so to find this angle it's times cosine of theta which is right is going to be the same thing it's times the length of which is always sine of theta so those are our relationships our trig relationships is what helps us to go back and forth between the two forms okay so polar coordinates are not unique okay if want to plot something in rectangular coordinates in and move over move up that's it there's only one point in rectangular coordinates that represents that point in polar coordinates you can have several points that represent the same point okay so the the form the representation of your point is not unique in fact you can have in infinite ways of representing really any one point okay so for example if want to express the point three negative three root three in polar coordinates four different ways okay so these would be the most four basic ways and then we can talk about how we would do it in an infinite number of ways right so the most basic thing that we're going to do is understand that this is my and this is my right i'm going to need and theta so from my and need to find which is the distance of this point from the origin okay and in order to find the angle will use tan inverse of over right so again if we come back to our formulas we're not using these formulas we are given point in and want to use my radius want to find my radius and my angle so this is the formulas i'm going to use okay okay so to find square both of these terms and add them together and then take the square root as we're used to doing so this is gonna be nine this is going to be nine times three okay so that'll give me 36. square root that is six okay here if want to find tan inverse of this value notice that these threes will cancel out so instead of writing just square root three can write it over one then can separate these two to be over the same denominator of two because that tells me nice angle that we have right so root three over two over one over two gives me pi over 3 as my angle okay now the problem with this is that yes this is the angle that's formed but look at the point right the point is in quadrant four okay because the value is negative value is positive so we have to adjust this angle so that it represents the correct angle value so we just have to subtract this from zero okay which is negative pi over three so we can write this as five pi over three if we want to talk about it in in positive angle form and so my first value first way of writing this would be six five pi over three so it has radius of six and an angle of five pi over three now in doing this found secondary angle that can use to represent this as well which is negative pi over three so the second one is already written for me would use radius of six and instead of five pi over three just use that negative angle right so really the way that can do this an infinite number of times is just find bunch of angles that represent negative pi over three and keep the radius of six positive and that would always land on that point okay now let's look at different way of doing this instead of having positive six let's say we use negative value of six okay well if use the negative value of six and negative pi over three then would point in the opposite direction of the angle that want right so i'd have to add pi to that in order to get me back into the correct direction right because remember pi is 180 that swings me back in the opposite direction so can use negative radius and adjust this by going 180 degrees back into the correct direction okay now obviously don't want to keep it as pi minus pi over three thought would do the math and would write this as negative six and two pi over three okay so would go in the two pi over three direction and then would go in the opposite direction for my radius to bring me to six five pi over three right so fourth way of doing that is the same thing that did before just find second keep negative six and find the secondary angle that gives me the same thing so instead of going in the positive counterclockwise direction would go in the negative clockwise direction so would go would take my pi over 3 and would add pi to that and then change its direction and that will give me negative 6 and negative four pi over three okay so just went in the negative direction and then changed this sign okay so that's four different representations for the same point now obviously some of these are harder to think of than the others number one is probably number one or two those are the ones you're going to want to stick with obviously we probably try to keep everything positive if we can so this is generally what you're going to be looking for for your answers okay but there are multiple ways to represent the same point in space when you are using polar coordinates so let's look at an example right let's find the rectangular coordinates instead for given set of polar coordinates so we're given polar coordinates of 2 7 pi over 6. if want to put these into rectangular coordinates again gotta understand that this is my my and my theta so got to use those equations that have is 2 my angle is 7 pi over 6. so in order to go in our other direction we're using this pair of equations right is cosine theta is sine theta when we're given polar and we're going into okay so is 2 angle 7 pi over 6 which is basically just pi over 6 right it's the same as negative pi over 6 basically so we can call this the cosine value here is square root 3 over 2 multiplied by 2 and those cancel and then this is going to be negative because it's in quadrant 3 right it's 1 pi over 6 over pi so it's in the negative side okay sine theta so that is also going to be negative value and it's going to have value of one half for 2 sine 7 pi over 6 but we're going to multiply that by 2 so that will actually give me negative 1. so my point here is going to be negative root 3 and negative 1 in rectangular coordinates so not only can we represent points in polar coordinates right but we can also take an equation that's already in our regular rectangular coordinates and we can turn it into polar equation basically that's what it's called okay so we use the same substitution right wherever see replace it with cosine theta wherever see replace it with sine theta if i'm going back the other way right let's say i'm given polar equation if want to turn it into rectangular equation use is square root of squared plus squared and that theta is tan inverse okay so use the same thing so no matter what whenever i'm going from one form to the other this is my key right for points for equations if i'm going from polar to rectangular always make these substitutions whether it's from points or for equations for equations or points if i'm going from rectangular into polar always use these substitutions okay so that's how we do that all right so okay so for an example let's convert equals squared into polar equation is always our dependent variable and theta is always our independent so we're always looking for equals something when we have polar equation okay it's always equals something so want to convert equals squared into polar equation so i'm going to use these substitutions is going to turn into sine theta is going to turn into our cosine theta so i'm going to make my substitutions my gets squared so i'm going to go ahead and square that so that turns into squared cosine squared theta and remember is my dependent variable so want to have equals something okay so i'm going to divide both sides by and then now can divide by cosine squared so can isolate okay so have equals cosine squared over sine squared and if want can simplify this to cosine theta cotangent theta okay either one of these is okay as an answer so my polar equations will always have trig functions in them okay so let's do one more example so let's go the opposite direction this time let's start in polar and let's try to find the equivalent rectangular equation right so my polar equation is equals 2 over 1 minus 2 cosine theta let's find its equivalent in rectangular coordinates so here's disclaimer lot of times we're not going to be able to find the equation as equals something sometimes we will but lot of times we want we won't and that's just the nature of polar coordinates polar coordinates has lot to do with circles right we're talking about angles and radii and things like that so if you think about what the equation of circle is right squared plus squared equals one that's an implicit equation right cannot explicitly say equals this right have to have an equation where is part of the equation and so that's typically what's going to happen not always we'll talk talk about the different forms what they look like in the next section but lot of times we are not going to be able to say equals this lot of times it's just going to be bunch of x's and y's together some squared some cubes some you know all kinds of stuff going on so if you can't get nice simple looking equation when you do this going from polar to rectangular that's fine you're probably on the right path actually okay so if have my equation okay am going to multiply both sides times 1 minus two cosine theta so it looks like this just so that's little bit easier for me to work with okay i'll go ahead and distribute this right and then want to start to make my substitutions so the reason did that was because here didn't have cosine had 2 cosine but when move it to the other side notice that now have and have an cosine theta cosine theta is the same as could not replace this for because it is not cosine theta it's 2 cosine theta it doesn't work that way have to have that exact term cosine theta to substitute my now know what is right it's square root of squared plus squared this is so make those two substitutions is equal to this is here so have squared plus squared minus two equals two what i'm going to do is can keep it like this this is technically correct although wouldn't really do math or plot this as is right want to get rid of this square root sign so i'm going to isolate this term and then by adding 2x to the other side and then i'll square root both sides okay so i'm going to add 2x to the other side of the equation i'm going to square this and square this and this is what would get out do not need to multiply this out right can keep it as is this is nice condensed form wouldn't have to do anything more than this if you wanted to multiply this out and then set the whole thing equal to zero you can do that if you did do that you might want to look at completing the square things like that try to get this to look like the equation of circle but as it stands this would be fine as an answer okay