in this video we're going to talk about how to convert rectangular equations into polar equations so what exactly is a rectangular equation and what are some examples of polar equations you need to be able to distinguish these two first a rectangular equation typically contains an x or y variable so x plus y equals 4 that's a rectangular equation x equals three y equals five or x squared plus y squared equals nine those are some examples of rectangular equations a polar equation has the variables r and theta so for example r equals four that's the radius of a circle that's a polar equation theta equals pi over three or r equals seven sine theta those are some examples of polar equations in this video our goal is to convert a rectangular equation into a polar equation and to do so there are some formulas that you need to know so let's draw a right triangle the hypotenuse of the right triangle is r which is the radius here's the 90 degree angle and this is theta so this is x and this is y so first you need to know that x squared plus y squared is equal to r squared and tangent theta is y divided by x and x is equal to r cosine theta and y is equal to r sine theta so go ahead and take a minute write these formulas down and then we're going to work on some examples so you can master your ability to convert a rectangular equation into a polar one so let's start with this example let's say that x is equal to eight go ahead and take a minute convert this rectangular equation into a polar equation now we know that x is equal to r cosine theta so that's the formula we want to use so let's replace x with r cosine theta and so that's going to equal 8. now we're going to do now is we're going to isolate r and let's divide cosine or let's divide both sides by cosine that's what i meant to say and so r is equal to 8 times 1 divided by cosine now you need to know that secant theta is equal to 1 over cosine by the way you could leave your answer like this 8 over cosine it should be fine or you can write it as a secant theta personally i prefer to write it like this but you could write it as 8 over cosine if you want to let's try another similar example let's say y is equal to 5. what's the polar form of this equation now we know that y is r sine theta so therefore r sine theta is equal to and now let's divide both sides by sine theta so r is five times one over sine theta and 1 over sine is cosecant so this is 5 cosecant theta and so that's the answer here's another example let's say that 5x plus 4y is equal to 8. go ahead and convert that into its polar form so as we mentioned before x is r cosine theta and y is our sine theta so our goal is to isolate r if we factor out r this is gcf we'll be left with 5 cosine theta plus 4 sine theta and that's going to equal eight so now what we need to do is divide both sides by five cosine plus four sine and so now this will give us our final answer so r is five divided by i mean not five but rather eight divided by 5 cosine theta plus 4 sine theta so that's all we can do for this particular example now what about this one let's say that x squared plus y squared is 16. what do we need to do first in this example so if you recall r squared is x squared plus y squared so therefore let's replace this with r squared so r squared is equal to 16. and now all we could do at this point is take the square root of both sides so r is equal to four so our goal is typically to solve for r we want to get r by itself on one side that's usually the case so that's it for this problem here's another one x minus 3 squared plus y squared let's say it's equal to 9. what's the first thing that we need to do in this example the first thing that we should do is we need to expand x minus 3 squared it's basically x minus 3 times x minus 3. now let's go ahead and foil it x times x is x squared and x times negative 3 that's negative 3x and this is also negative 3x and then negative 3 times negative 3 that's positive 9. so if we subtract both sides by 9 these numbers will cancel and we can also combine like terms so we now have is x squared i'm going to write these two together x squared plus y squared minus six x is equal to zero now if you recall x squared plus y squared is equal to r squared so we have r squared minus six and x is r cosine theta so that's equal to zero so now let's take out an r if we factor out r we'll be left with r minus six cosine theta since r and r minus six cosine theta if you multiply them if they equal to zero any one of them can be zero zero times anything is zero so we could set this portion r equal to zero and we could set r minus six cosine theta equal to zero now for this one we need to add six cosine to both sides so this gives us two possible answers r is equal to six cosine theta or r is equal to zero here's another problem let's say that x squared plus y plus four squared is equal to 16. so based on the last example feel free to pause the video and try this one see if you can get this a similar answer to the last one so just like before we need to foil y plus 4 squared so y times y that's going to be y squared and then y times 4 is 4y and then 4 times y that's also 4y and then 4 times 4 is 16. now we know that x squared plus y squared we can replace that with r squared next we can add 4y plus 4y which adds up to 8y and if we subtract both sides by 16 we're going to have r squared plus 8y is equal to 0. now let's replace y with r sine theta and let's take out an r so this is going to be r plus a sine theta and that's equal to zero so now we could set r equal to zero and r plus eight sine theta is equal to zero so this is the first answer and for the second answer r is equal to negative eight sine theta and so that's it what if y squared is equal to 4x what can we do to convert this equation into its polar form so first we know that y is r sine theta but since we have y squared this is going to be r sine theta squared and x is r cosine theta so what we have now is r squared sine squared theta is equal to 4 r cosine theta now if we divide both sides by r at this point we're going to lose an answer and that answer is r is equal to zero which sometimes some textbooks just completely omit that answer but i'm going to get both answers so i'm going to subtract both sides by 4r cosine so basically i'm going to take this move it to the left it's positive on the right side but it's going to be negative on the left side so i'm going to factor out the gcf which is r and i'll be left with r sine squared theta minus four cosine theta so my first answer r is equal to zero and then the other answer which is the one that the textbook usually wants it's r sine squared theta minus four cosine theta is equal to zero so we're going to focus on that equation so i'm going to add 4 to both sides r sine squared theta is now equal to 4 cosine theta so let's divide both sides by sine squared at this point so r is four cosine theta divided by sine squared so you can leave your answer like this if you want to but do know that you can change it also for example we can expand sine squared into sine times sine i'm going to write it as cosine over sine times one over sine cosine divided by sine is cotangent and one over sine is cosecant so you could write your answer like this four cotangent cosecant so both answers are acceptable let's work on another similar example let's say that x squared is equal to 8y try that problem so we're going to replace x with r cosine theta let's not forget to square it and let's replace y with r sine theta so r squared cosine squared is equal to eight r sine theta now in the last problem we took the term on the right and move it to the left we're not going to do that in this example we're just going to divide both sides by r just keep in mind when you do this you lose an answer and that answer you lose is r equals 0 which most textbooks are going to ignore that answer anyway but just want you to keep that in mind so we're going to have one r left over on the left side so r cosine squared is equal to eight sine theta and now let's divide both sides by cosine squared now we have the answer r is equal to 8 sine theta divided by cosine squared but now let's change it so let's rewrite it as sine divided by cosine times one over cosine now sine over cosine is equal to what other trig function sine divided by cosine is tangent theta and one over cosine is secant theta so therefore r is eight tangent theta secant theta now what about this problem x squared plus y squared let's say it's equal to six x plus four y try that so we know that x squared plus y squared that's equal to r squared and x is r cosine theta y we know it's r sine theta so r squared is equal to six r sine theta i mean not sine but cosine theta plus four r sine theta now what do you think we should do at this point at this point what we need to do is divide each term by r and keep in mind we'll lose the answer r equals zero so you can just add it towards the end so right now we have r is equal to six cosine theta plus four sine theta so that's the equation that we want but keep in mind that r can be zero because zero squared and six zero squared is equal to six times zero times cosine plus four times zero times sine because if this is equal to zero and that's equal to zero then this will be equal to zero as well so zero will make that equation true but the answer that we want is this one r is equal to six cosine theta plus four sine theta now what about this one let's say y is equal to the square root of 3 times x let's convert that equation into its polar form so y is equal to r sine theta and x is our cosine theta so if we divide both sides by r these will cancel and we're not going to have anything in terms of r so one possible answer is r is equal to zero that would make this equation true both sides will be equal to zero but let's not worry about that answer right now we have sine theta is equal to root 3 times cosine theta and let's divide both sides by cosine so sine divided by cosine is equal to the square root of three and if you recall sine divided by cosine is tangent so tan theta equals root three so we're going to do is take the arctan of both sides the arc tangent of tangent these two will basically cancel and on the left we're just going to have the angle theta so the angle theta is equal to the arctan of root 3. you can either type this in your calculator if you do make sure it's in radian mode but arc tangent of root three is pi over three or tangent pi over three is the square root of three and so basically we just get an angle so that's the answer you