 Welcome back everyone In this video, we're going to talk about how you can convert from polar coordinates to Cartesian coordinates and vice versa So imagine we have a point P which is in the plane And if we know it's if we know it's Cartesian coordinates x comma y what that means for us is the following We know that the distance that the point lives above the x-axis that distance is y And we know that the distance that lives to the right of the y-axis is x and this will form a right angle right here And that's exactly what Cartesian coordinates mean now on the hand polar coordinates want to measure the distance from the origin to this point we call that r and Then the angle formed between this red line with the x-axis that angle we call theta And so these four coordinates x y r and theta Satisfies some trigonometric relationships to each other and so basically get the so katala relationships. You're going to get that X over r equals cosine of theta If you cleared the denominators you get the following equation right here That's the ca from so katala if you take y over r This is equal sine theta you get opposite over hypotenuse cleared the denominators You get this equation right here and we just pause there for a moment This actually gives us a way of converting from polar coordinates So you know r and theta we can convert those over into Cartesian coordinates x and y in the following way This equation the right hand side only depends on r and theta so you can compute an x and you can compute a y Again, this is just this is just trigonometry here And so consider for example the polar coordinate 2 comma pi thirds. This is polar We want it to be Cartesian what this tells us is we know r which is 2 we know theta Which is pi thirds and we want to find x and y so x will equal r cosine r Cosine theta which means it's equal to 2 times cosine of pi thirds pi thirds is equal to 1 half and So you get 2 times 1 half the x coordinates of 1 to find y you're going to use r sine theta Which this would be 2 times sine of pi thirds Sine of pi thirds is root 3 over 2 the 2 is canceling you get the square root of 3 and So then the the Cartesian coordinate associated to the polar coordinate 2 comma pi thirds will be 1 comma the square root of 3 And that's all one has to do to switch from polar coordinates to Cartesian coordinates coming back here to go from To go from Cartesian coordinates x comma y into polar coordinates r comma theta You're going to use the remaining statements about this right triangle that we were talking about over here So for example, we've done so calm. We need the toa We get that y over x is equal to tangent theta Using this equation right here notice if we know x and y over here Then we can compute theta by taking arc tangent if necessary, right? I do prefer this equation right here tangent theta equals y over x because they're actually infinitely many solutions to this equation tangent theta equals y over x this will involve Co-terminal and supplementary angles and that's because there is more than one way One more than one angle to represent a polar coordinate this equation has that Multiplicity built into it But you can certainly just accept arc tangent of y over x and go with what your calculator tells you and then to find out Are we use the Pythagorean equation x square plus y squared equals r square? That is the adjacent side square plus the opposite square side squared equals the hypotenuse squared We get the Pythagorean equation right here you could Take the square root of both sides r equals the square root of x squared plus y squared But really there's a plus or minus that's associated to that and that's again from the fact that r could be positive It could be negative for the most part. We'll just take the positive one and So if we have the Cartesian coordinate one comma negative one and we want it to be polar What this tells us is that x equals one y equals negative one So r squared will equal the school will equal one square plus negative one squared That is one plus one which equals two so we'd say r equals a square of two Tangent of theta Equals negative one over one which is negative one and so then arc tangent theta will equal arc tangent Of negative one which equals negative pi force So we could describe this polar coordinate as square root of two comma negative pi force If you want a positive angle you could just add two pi there in which case you'll get root two comma seven pi over four And that gives you a polar representation of that Cartesian coordinate There are other ways of representing it, but we don't need every representation. We just need one to describe this polar coordinate