in this video we're going to talk about how to convert a polar equation into a rectangular equation so here are some formulas that you need to know let's consider this a right triangle this is x this is y this is r and here's the angle theta so you need to know that x squared plus y squared is equal to r squared and x is r cosine theta y is r sine theta and tangent theta is y divided by x so these are some formulas that you'll need in order to convert a polar equation into a rectangular equation so what is the difference between a polar equation and a rectangular equation how do we distinguish the two so here's an example of a polar equation r is equal to five sine theta a polar equation would have the variables r or theta a rectangular equation on the other hand consists of x and y variables so for example x squared plus y squared is equal to four that's a rectangular equation another example is let's say r is equal to a constant seven that's a polar equation or x is equal to three that's a rectangular equation another one you might see is theta let's say it's equal to pi over four compared to x squared is equal to 4y so rectangular equations have the variables x and y polar equations have the variables r and theta so let's work on an example let's say r is equal to seven so convert this polar equation into a rectangular equation so what do you think one needs to do in this particular example for each of these examples pause the video and try it yourself and then unpause it to see the solution the best thing to do is take the square of both sides r squared is equal to 7 squared which is 49 and if you recall x squared plus y squared is equal to r squared so therefore x squared plus y squared is equal to 49 and that's all we need to do for this example so that's how you can convert that particular polar equation into a rectangular equation here's another example let's say r is equal to 5. for the sake of practice go ahead and try so we know that r squared has to be 25 and x squared plus y squared is equal to r squared so that's equal to 25. so anytime r is equal to a constant number that's what you need to do now what if we have the angle theta let's say it's equal to pi over four convert that equation into a rectangular equation now the first thing i would recommend doing is to take the tangent of both sides so tangent theta is equal to tan pi over 4 and as you recall tan theta is y divided by x so that's one form you want to keep in mind so on the left side we have y divided by x and tan pi over four is one so all we need to do is multiply both sides by x so therefore in this example y is equal to x now how about this one let's say if theta is equal to zero degrees try that problem so just like before we're going to take the tangent of both sides so tangent theta is equal to tangent of zero degrees and we know tan theta is y divided by x now what's tangent of zero degrees tan zero is zero now if we multiply both sides by x these two will cancel and so y will equal zero zero divided by anything is zero so zero over eight is zero so it really doesn't matter what is the value of x if y over x is equal to zero then y has to be zero so this is the answer now what about this example let's say if theta is pi over 2 try that one so we need to take the tangent of both sides so tan theta is equal to tan pi over two and tan theta is y divided by x now tangent pi over two is undefined so if tangent pi over two is undefined what do you do in this case under what circumstances is a function undefined a function is undefined if you have a zero on the bottom if you have a zero on top the whole thing is zero that's why the last example y was equal to zero so if y over x is equal to zero y has to be zero but let's say if you have seven over zero this is undefined or three over zero that's also undefined so therefore for something to be undefined it's going to be some number n divided by zero so we could say that x has to be zero it doesn't really matter what y is equal to as long as x is zero the whole thing will be undefined so therefore the answer for that problem is x is equal to zero let's say if r sine theta is equal to five and also r cosine theta is equal to 4. convert each polar equation into a rectangular equation so if you recall y is equal to r sine theta so therefore we can replace r sine theta with y and this will give us the answer so y is equal to 5 and that's all you got to do now for the other one it turns out x is equal to r cosine theta so we just got to replace r cosine theta with x so the answer is x is equal to four and so that's it for this problem how about this example let's say r is equal to 3 cosecant theta convert it into a rectangle equation so cosecant is equal to what other function it turns out that cosecant is 1 divided by sine theta so what we need to do is multiply both sides by sine theta by doing so we can get rid of the sine theta on the right side so what we now have is r sine theta is equal to three and based on the last example we know that y is equal to r sine theta so therefore y is equal to three and so that's the answer for this problem here's another similar question that you could practice on try this one let's say r is equal to 4 secant theta so secant is 1 divided by cosine theta so now let's multiply both sides by cosine theta so these two will cancel r cosine theta is equal to four now we know that x is equal to r cosine theta so therefore x is equal to four and so that's it let's say if r is equal to three sine theta try this one what do you think we need to do in this problem what i would recommend doing is multiplying both sides by r whatever you do to the left side you must also do to the right side so now what we have is r squared is equal to 3 r sine theta now if you recall x squared plus y squared is equal to r squared so we're going to replace r squared with x squared plus y squared also we know that y is equal to r sine theta so we're going to replace r sine theta with y so 3 3r sine theta is equal to 3y and this is the equation x squared plus y squared is equal to 3y so for the sake of practice try this one let's say r is equal to 4 cosine theta so just like before we're going to multiply both sides by r so r squared is 4 r cosine theta and r squared we know it's x squared plus y squared and r cosine theta is x so 4 r cosine theta is 4x and so this is the answer x squared plus y squared is equal to 4x try this one let's say if r is 3 cosine theta plus 5 sine theta convert it into its rectangular form so just like we did before we're going to multiply everything by r so on the left side we're going to have r squared on the right side we're going to have 3 r cosine theta and then plus 5 r sine theta so we know that r squared is x squared plus y squared and r cosine theta is x so this is going to be 3x r sine theta that's equal to y so what we're going to have is 5 y and so that's the answer x squared plus y squared is equal to 3x plus 5y let's try this one let's say if r is equal to 5 divided by 2 cosine theta plus 3 sine theta what do you think we need to do for this problem what i would recommend doing is multiplying both sides by the denominator of this fraction that is by 2 cosine theta plus 3 sine theta so on the right side these two will cancel but on the left side we're going to get r times 2 cosine theta plus three sine theta and so all of that will be equal to five now what we need to do is distribute r so what we're gonna have is uh two r cosine and then 3 r sine and that's equal to 5. as you know r cosine theta is x so what we have is 2x and our sine theta is equal to y so this is 3y so we have a linear equation in standard form 2x plus 3y is equal to 5 and that's the answer let's try this one let's say r squared sine two theta is equal to eight how can we change r and theta into x and y for this particular example the first thing you want to use is the double angle formula sine 2 theta is equal to 2 sine theta cosine theta so this is going to be r squared times 2 sine theta cosine theta next let's divide both sides by 2. so we can get rid of this number so what we now have is r squared sine theta cosine theta is equal to eight divided by two which is four r squared we can rewrite that as r times r so i'm going to separate r sine theta and also r cosine theta r sine theta is equal to y and r cosine theta is equal to x so y times x is equal to four if y x is equal to 4 then let's get y by itself so therefore y is equal to 4 divided by x so we have a rational function and that's the answer what if r is equal to 8 divided by cosine theta what's the answer in this example what we need to do is multiply both sides by cosine and we know that r cosine theta is x so x is equal to eight so that problem wasn't too difficult now what about this one let's say if r is equal to five cosine theta divided by sine squared what do you think we need to do for this problem so feel free to pause the video and try now what i would do is multiply both sides not by sine squared but by sine theta and you'll see why so on the left i have r sine theta on the right i have 5 cosine theta now i have 2 sine thetas on the bottom because it's sine squared only one will cancel which will leave one left over so what we have is r sine theta is equal to 5 times cosine divided by sine now we know that tangent theta is y divided by x which means that cotangent theta which is 1 divided by tangent theta is x over y now cosine over sine is cotangent theta and r sine theta we know it's y so y is equal to 5 cotangent theta and cotangent theta is x divided by y so now all we need to do is multiply both sides by y and now we have the final answer y squared is equal to 5x and so that's the solution let's try one more example let's say r is equal to sine theta cosine squared theta let's convert that into its rectangular form so what do you recommend that we should do in the first step the best thing to do at least what i think to do is to multiply both sides by r cubed so on the left this is going to be r to the fourth on the right r cubed sine theta cosine squared now r to the fourth i'm going to write it as r squared squared r cubed i'm going to expand it as r times r times r and cosine squared i'm going to expand it and write it as cosine theta times cosine theta now i'm going to put these two together so that's going to be r sine theta and then let's put these two together so that's our cosine theta and let's do the same with the last two that's r cosine as well now we know r squared is x squared plus y squared and all of that is raised to the second power r sine theta is equal to y r cosine theta is equal to x so what we now have is x squared plus y squared squared is equal to y x squared now let's take the square root of both sides so the square root will cancel with the square on the left we're just going to have x squared plus y squared on the right we have the square root of y times the square root of x squared if we expand it the square root of x squared is x and we can't simplify root y so this is it x squared plus y squared is equal to x root y that's the final answer you