X prime of theta, derivative of the first expression, it is going to be two Fair enough and we couldĭo the same thing for X. The second expression, derivative of the sin of That's just coming out of the chain rule and then times the second expression, sin of theta and then plus, plus the first expression, sin of two theta, sin of two theta times the derivative of Rule right over here, derivative of this first expression is two cosine of two theta, cosine of two theta, we've already seen that. Well, this is once again, we're just gonna use ourĭerivative techniques, so I could write Y prime of theta, the derivative of Y with respect to theta, just gonna use the product Of Y with respect to theta, find a general expression for it. So, Y would be equal to sin of two theta, sin of two theta times sin of theta, times sin of theta and X is going to beĮqual to sin of two theta, sin of two theta times cosine of theta, times cosine of theta, just like that but now we can use these expressions to find the rate of change How do we do that? Well, we know that R isĮqual to sin of two theta, so you just have to replace these Rs with sin of two theta. But now we can use these to express purely in terms of theta. To Y over our hypotenuse which is R and cosine of theta is equal to the adjacent or X over R and you just have multiply both sides of these equations by R to get to what we have right over there and once again, if this is going too fast, this is a review of just polar coordinatesįrom pre-calculus. Of our trig functions, sin of theta is opposite over hypotenuse, sin of theta is equal
![graph polar coordinates graph polar coordinates](https://mathbooks.unl.edu/Calculus/images/imagesChap15/polargrid.png)
Well, we know from trigonometry from our unit circle definition, the SOHCAHTOA definition Well, the height of that side is going to by our Y and then the length of this side is going to be our X. 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.
![graph polar coordinates graph polar coordinates](https://useruploads.socratic.org/2o1zhcJwSVif1lxqGRMx_desmos-graph.png)
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
![graph polar coordinates graph polar coordinates](https://cdn.free-printable-paper.com/images/large/polar-graph-paper.png)
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 \) will yield the maximum \(| r |\).Have here is the graph of R is equal to sin of two