These angle R combinations right over here, so let's say this is Now, just as a really quick primer, why does that make sense? Well, let's just take one of Say rectangular world, you have to remember the transformation that Y is equal to R sin of theta and that X is equal to R cosine of theta. So, one primer, a review from pre-calculus is that when you wanna goīetween the polar world and the, I guess you can Alright, that was interesting but let's see if we can express this curve in terms of Xs and Ys and then think about those derivatives. Is going to be cosine of two theta and then you multiply that, times the derivative of two theta with respect to theta which is two, so we could just say times two here or we could write a two out front. So, the derivative of sin of two theta with respect to two theta Take the derivative with respect to theta right over here. You just have one variableĪs a function of another. What is R prime of theta? Well, there's really nothing new here. The rate of change of R with respect to theta? Pause this video and see This in a calculus context, so the first question might be well, how do we express So, we reach a kind of a maximum R there and then and as theta increases, our R once again starts to get smaller and smaller and smaller. Over four right over there, well, sin of two times pi over four is sin of pi over two, R is equal to one. Is equal to pi over four? When theta is equal to pi Or clover-looking thing, so it starts looking like that and we could keep going all the way. R is going to be zero, sin of two times zero is just zero, so our R we're just gonna be at the origin and then as theta get larger, our R gets larger and so, we start tracing out this pedal of this flower And just to familiarize ourselves with this curve let's just see why it's intuitive.

Point right over here and specify it with some angle theta and some R which is the distance from the origin to that point. Them in terms of an angle and a radius so for example, this would have some X coordinate and some Y coordinate or we could draw a line from the origin to that So, what we're doing for any point here, we could obviously specify these points in terms of X and Y coordinates but we could also specify Let's just familiarize ourself why this graph looks the way it does. Our pre-calculus section but I'll give you a littleīit of a primer here. Search for polar coordinates in Khan Academy or look at Look unfamiliar to you or if you need to brush up on them I encourage you to do a Theta in polar coordinates and if polar coordinates Have here is the graph of R is equal to sin of two

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