in this video we're going to talk about how to graph polar equations these include circles limassons rose curves and laminas gates so let's start with a circle the first equation you may see is r is equal to a cosine theta now if a is positive this is going to be a circle directed towards the right now granted my circle is not perfect so bear with me a is basically the diameter of the circle and this is going to be the center of the circle so if you go up to find this point here it's half of a so if a is greater than 0 if a is positive it's going to open towards the right and if a is negative you're going to get a circle let me draw a good looking circle this time you can get a circle that is directed on the left so keep in mind this is going to be a and this is half of a so let's try some examples let's say if we have the graph r is equal to 4 cosine theta if you want to feel free to pause the video and try it yourself so we're gonna have a circle on the right side now a is four and half of a well four divided by two is two so that's half of a so what we're to do is travel 4 units to the right and then up 2 units and down 2 units so the circle is going to start at the origin and it ends at 4 on the x-axis from the center which is at two we need to go up two units and down two units and then simply just connect it so that's how you can graph r equals four cosine theta let's try another one let's try this one let's say that r is equal to negative six sine theta i mean that's sine but cosine beta we can get into sine later now because a is negative the circle is going to be on the left side on the x-axis so let's travel 6 units to the right since a is negative six one half of a is negative three now don't worry about the negative sign too much negative sign just tells you if the circle opens to the left now the center is going to be at negative three which is here so we need to go up three units and down three units so the graph is going to be at the origin at negative six at negative three three and at negative 3 negative 3. and so that's how you can plot the circle so keep in mind this is equal to a that distance and this is also equal to a as well which means this part is one half of a so half of a is basically the radius of the circle so if for some reason you need to find the area of the circle you can use this equation pi r squared the radius is 3 so it's pi times 3 squared which is 9 pi so now the next form we need to know is r equals a sine theta cosine is associated with the x values so as you can see the circle was associated with the x axis sine is associated with the y values and so the circle is going to be centered on the y axis so let's say if a is positive then we're going to have a circle that goes above the x-axis centered on the y axis so once again the diameter will still be equal to a and this portion the radius is half of a so that's when a is positive or when a is greater than zero now in the other case if a is negative or if a is less than zero the circle is still going to be centered around the y axis but it's going to open in a negative y direction so it's going to be below the x-axis and so the radius as you mentioned before is just one-half of a and the diameter is equal to a so let's try some examples let's say if r is 2 sine theta go ahead and graph that so first we need to travel up two units a is two half of a which is the radius is one so we're going to have these two points now let's travel one unit to the right and one unit to the left so the green dot is the center of the circle so we're going to travel one unit to the right and one unit left from it and so this is going to be the graph try this one let's say r is negative 8 sine theta go ahead and work on that example now the majority of the graph will be below the x-axis so i'm going to focus on that so let's travel eight units down and then half of a or four units to the right and 4 units to left so the point is going to be at the origin and 8 units down the center is 4 units down so if a is 8 the radius is one half of a which is four so we gotta travel four units to the right from the center and four to the left and so this is going to be the graph so now you know how to graph circles when you're given a polar equation now the next type of graph that we need to go over is the lima song and the equation is r is equal to a plus or minus b sine theta now if you have positive sign it opens towards the positive y axis that is the upward direction negative side opens in a downward direction in the negative y direction you could also have a plus or minus b cosine theta so if cosine is positive it's going to open towards the right in the positive x-axis direction and if cosine is negative it's going to open towards the left so let's draw the general shape if it opens towards the right so this is the lima song with the inner loop and you get this particular shape if a divided by b is less than one now both a and b represent positive numbers a and b are both greater than zero so if you get the graph three minus four sine theta b is not negative four b is positive four and a is positive three so let's say if it was three plus four cosine theta both a and b would still be three and four positive three and positive four so a and b are not negative now the next shape that we have if a divided by b is equal to one is the heart shape limassol also known as the cardioid and here's the generic shape for it so it has like this uh this dimple so it looks something like that maybe i could draw that better so that's the cardioid now the next one is the dimpled limousine with no inner loop so that occurs if a over b is between 1 and 2. so let's start with the x-axis it's a small dimple sometimes it's hard to notice so that's the dimpled lima saw with no inner loop the next one need to know is if a divided by b is equal to or greater than two so this limo song looks almost like a circle but it's not there's no dimple and there's no inner loop so i'm going to start from the left i'm going to draw it straight up and then looks like this but it's not exactly a circle because as you can see the right side is like bigger than the left but it almost looks like a circle so that's the uh the lima song without a dimple or an inner loop so those are the four shapes you need to be familiar with let's graph this equation let's say r is equal to three plus five cosine what do you think we need to do here we know this is a type of lemur song it's in the form a plus or minus b cosine theta so first we need to identify a and b a is equal to 3 and b is equal to 5. now we need to see if a over b if it's less than one if it's between one and two if it's equal to one or greater than or equal to two so a over b that's three over five and three over five as a decimal is 0.6 which is less than one now because it's less than one we know we have the limousine with the inner loop now there's four types the first type is if it's positive cosine this graph will open towards the right the next type is if we have negative cosine and in that case this graph would open towards the left if it's positive side then it's going to open in the positive y direction and if we have negative sign it's going to open towards the negative y direction so it's going to look something like that so just keep that in mind that's the first thing that you look for so we have positive cosine which means it should open towards the right side now when graphing this type of limousine you want to make sure you get four points two x-intercepts and two y-intercepts so let's uh draw a rough sketch of this graph this point is actually positive a it's a units relative to the center and this other y-intercept is negative a units from the center the first x-intercept which is associated with the inner loop it's the difference between a minus b so it's the absolute value difference of a minus b or you just say it's b minus a because b is going to be bigger now the second intercept is the sum of a and b and that's all you need to get a good decent graph if you can plot those four intercepts then you should be fine so let's go ahead and do that so in this case we can see that a is equal to three so we need to go up three units and down three units so those are the y intercepts now b minus a that's going to give us the first intercept that's five minus three that's two so here's the first intercept and then uh a plus b that will give us the second intercept that's three plus five which is eight so that's how you could find the two x intercepts now let's go ahead and graph it so first let's start with the inner loop and then let's go towards the first y intercept and then the second x-intercept and then towards the other y intercept so that's a rough sketch of this graph so the points that you need is uh 3 and negative 3 on the y axis and 2 and 8 on the x axis let's try another example so let's say r is equal to two minus five sine theta so try this one the first thing i would keep in mind is what type of in what direction will it open we know that a over b which is two over five it's less than one so this is a lima sum with an inner loop but notice that we have a negative sign so therefore it has to open in a negative y direction so we can see that a is 2 that's going to give us the y intercepts and b is 5. so let's go ahead and graph it well in this case because it opens downward a is actually going to be associated with the x-intercepts this time instead of the y intercepts so it's going to switch roles so we need to travel two units to the right and two to the left if we're dealing with cosine then a would be associated with the y-intercepts but because we're dealing with sine the rules are reversed now a plus b that's going to be 2 plus 5 that's 7 and b minus a 5 minus 2 is 3. so we're going to travel 3 units down and also 7 units so we're going to have two y-intercepts so let's start with the inner loop and then let's let me do that again and then let's get the x-intercept and that's it so make sure you get these two x intercepts negative two and two and the y intercepts negative three negative seven and as you mentioned before because we have negative sign it has to open in a negative y direction now let's try this problem let's say that r is 3 minus 7 cosine theta so go ahead and pause the video and work on that example so let's find the value of a over b a is three b is seven and three over seven is less than one because seven over seven is one so what we have is the inner loop limousine now we're dealing with negative cosine which means it's going to open towards the left so the majority of the graph is going to be on the left side now a is associated with the y-intercepts when dealing with cosine when dealing with sine as we saw in the last example a is associated with the x-intercepts so we're going to travel three units up and three units down to get the y-intercepts when dealing with cosine for sine you need to know that a is associated with the x-intercepts so next let's find the first x-intercept which is going to be b minus a or seven minus three and that's four and then a plus b three plus seven is ten so because we're going towards the left we need to travel four units to the left that's going to give us the first x-intercept and then 10 units to the left relative to the origin so that's the second x-intercept now let's go ahead and graph it so let's start with the origin and let's draw the first inner loop and then let's focus on the outer loop and so that's it that's how you can graph it my graph is not perfect but at least that's the general shape you get the picture now let's try this one let's say that r is three plus three cosine theta what do we need to do here well first we need to determine what type of limousine we have a and b are the same when a and b are the same a over b is equal to one and in that situation we have the heart shaped lima star also known as the cardioid and because cosine is positive it's going to open towards the right now i'm just going to draw the general shape of the cardioid which it looks like this so there is no inner loop now when dealing with cosine the x and y intercepts are going to be a again a and negative a now the first x-intercept is just going to be the origin and so we don't have to do anything it's just going to start from the origin now the second x-intercept is a plus b we don't have the inner loop which was a minus b for the inner loop limit sum so we don't have to worry about a minus b or b minus a so now let's go ahead and graph it so in this example a is string so we're going to go up three units and down three units and a plus b that's three plus three so that's equal to six so the x-intercept is going to be six comma zero and a y intercepts are zero three and zero negative three so now that's we're gonna have to graph like this and that's it that's how you can graph the heart-shaped limousine you