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# Polar coordinates

## Video transcript

everything we've done up to now has dealt with Cartesian coordinates even though you might have not realized that they were Cartesian coordinates because I never called it that before just now so what is a Cartesian coordinate let me draw the axes that you're hopefully familiar with by now if you're not review those videos let me just draw the y axis and the x axis and so cartesian coordinates and actually they can apply to more than just two dimensions but two dimensions is what we tend to deal with and then if I want to specify any point in two-dimensional space I just tell you how far kind of in the X direction I have to go and how far in the Y direction and I can specify that point like let's see if I said well let me give the Cartesian coordinates for that point right there and I might say well to get to that point right there I have to go to the right three steps so this is three right here and then I have to go up four steps one two three four and you end up there and by convention we call this the x coordinate we call this the y coordinate and we call this three comma four and this is just a convention where we say the first coordinate is how far in the X direction we go and the second coordinate is how far in the up-and-down or in the Y direction we go since three comma four and we could have gone three to the right and four up or we could have gone four up and three to the right we've gotten to the exact same point this is just one way to really specify a point in two dimensions another option and you might have done this in everyday life is to say hey let me just point you in a direction and then you go a certain distance in that direction so I could have said you know what let me just point you in this direction let me just point you in that direction you need to go this far and you need to go this far in that direction and that would also give you the same point so how do i specify the direction well how about I call this zero degrees how about I call this the x-axis how about I call that zero degrees and we'll deal with degrees but hopefully you know how to convert between degrees and radians so you could just as easily convert to radians but you could specify the direction if you call this zero saying okay this is theta degrees and I want you I'm going to point you in a degree and you could almost imagine you know someone who's blindfolded you point them in this direction and you say walk our units and then they'll end up at that same point and so you could also specify this point instead of specifying this is kind of the Cartesian coordinates this is X comma Y you could also specify it maybe if we can figure out a way to do it and I'll do it in magenta you could also specify it as R comma theta which essentially says you know walk our units in the theta direction let me act let's figure what that out is because I think it's kind of abstract now it might be a little difficult to understand so we're going to break out a little trigonometry and actually just a little Pythagorean theorem can we figure out R and theta well R hopefully is is is the easier one here because if you think about it this is a right triangle right this is a right triangle that has distance three this has distance 4 this is a right triangle so what is R here Pythagorean theorem 3 squared plus 4 squared is equal to hypotenuse squared or R squared so we could say 3 squared plus 4 squared is equal to R squared that's 9 plus 16 is equal to R squared 25 is equal to R squared we don't want to deal with negative distances so 5 is equal to R R is equal to 5 fair enough so we already figured out that R is equal to 5 how do we figure out theta so what do we know here we know we're trying to figure out theta and we know it's opposite side all right this is just going back to sohcahtoa right let me write down sohcahtoa so kanto if this is completely unfamiliar to you you might want to watch the basic trigonometry vo so we know the opposite side of theta right that's 4 and we also know the adjacent side which is 3 so which which of the trig functions uses the opposite and the adjacent well the Toa opposite and adjacent so tangent so tangent of theta tangent of theta is equal to the opposite which is really the y-coordinate which is equal to 4 over the adjacent which is the X which is three so tangent of theta is equal to 4/3 and to solve this you essentially just take the inverse tangent of both sides of this so this is the same thing and depending on your calculator or the convention you use you might write this when you take the inverse tangent of both sides you get well I'll just write it out you could write it as the arc tan arc tan of the tangent of theta is equal to the arc tan of 4/3 and this of course the arc tan of the tangent this is the same thing as the inverse tan of the tangent of theta this is just equal to theta that just simplifies to theta is equal to arc tan of 4 over 3 and then another way to write arc tan is often they'll often write it I'll write it so the same exact statement could be written as tan inverse to the negative 1 power they sometimes say it although it's a little it's a little you know it's a little misleading sometimes when they write it like this because you don't know am I taking the whole tangent of theta to the negative power or something but sometimes I write arc tan is 10 to the negative 1 power but either way so we can figure out theta by taking the arctangent of 4/3 and most people do not have the arctangent of 4/3 memorized and I don't know an easy way to figure it out without getting my ti-85 out so let's get it out all right so how do we know the arc test we want to do this inverse let me get the calculator up a little higher as you can see we want to get this inverse tan right here if you press 2nd inverse tangent of 4/3 and I already set my calculator degree mode it gives you 53 point one three degrees so theta is equal to 53 point one three degrees so there you have it we already knew that we could specify this point in the two-dimensional plane by the point X is equal to 3 y is equal to 4 we can also specify it by R is equal to 5 and theta is equal to 53 degrees so I'll write that so it could also in polar coordinates it can be specified as R is equal to 5 and theta is 53 point 1 3 degrees so all that says is okay orient yourself 53.3 you know fifty three point one three degrees clot up was this counterclockwise from the x-axis and then walk five units and you'll get to the exact same point and that's all polar coordinates are telling you let's do another one actually before that let's see if we can come up with something general here because once you have the general then then you can always solve for the particular so let me just write an arbitrary point last time I gave a specific point the point was it three comma four I think it was so let's say I have the point x comma Y X comma Y it's right there so its x-coordinate is there its y-coordinate is there so how do we convert this to our two polar coordinates R theta well let's do the same thing we did in the last video see if this is this length is R and this angle is Theta so just going into what we did in the last video we use Pythagorean theorem we say that x squared plus one because this distance here is also Y right this distance here is X Pythagorean theorem you get x squared plus y squared is equal to R squared that's just the Pythagorean theorem and then you have the tangent of theta the tangent of this angle tangent of theta sohcahtoa Toa opposite over adjacent tangent of theta is equal to the opposite which is y over the adjacent which is X and this is really all you need and if you wanted to go one step further and this is actually something that you could you should kind of right at the side of your paper how can we express Y is a function if we wanted to go the other way if we were given R and theta how could we get Y well let's think about it Y if we want to know see if we have if we've given our and Y and we want to figure out theta or you know what deals with the R and the y with respect to theta R is a hypotenuse and Y is the opposite let me write sohcahtoa down so Toa so what deals with the opposite and the hypotenuse opposite and a hypotenuse sign so the sign of theta the sine of theta is equal to the opposite which is the y side which is equal to Y over the hypotenuse which is R equal to R and so if you multiply both sides by r you get what are sine theta is equal to Y and then let's do it the other way what if we wanted to have an equation for theta R and X X is adjacent to the angle R is tilde hypotenuse so what deals with adjacent and hypotenuse well that's cosine so we have the cosine of theta is equal to the adjacent is equal to x over the hypotenuse which is R multiply both sides by r and you get X is equal to R cosine theta and really when you're equipped with this formula which really just comes from our trig identities this formula which you know similar just comes from sohcahtoa this which comes from sohcahtoa and really is you know you if you're given these first if you're given these two you can easily derive that one and this one which is just the Pythagorean theorem you're actually ready you're fully equipped to convert between polar and rectangular coordinates and we'll do that in the next video because I just realized I'm out of time