In Exercises 13–34, test for symmetry and then graph each pola...

Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

Accepted Answer

Test the polar equation for symmetry and sketches graph R equals six minus six sine theta. But we're also given a polar graph here to try and plot our equation on. Now to solve this, we need to check three types of symmetry. For this equation. Our first type of symmetry is polar symmetry. For polar symmetry, we need to check if we have the angle data change the negative data. Do we get the same equation back out? So we will replace our data with negative data. We get R equals six minus six sine negative data. Now sign is actually an odd function which this means if we have sun of an angle negative data, we can change this into negative sign theater. So our function here will replace R equals six minus six, multiplied by negative sine theta. We then get R equal six plus six sine data. When distributing our negative sign. This is not the same as our original function. So we cannot say for sure if this has polar symmetry, it most likely doesn't, but we can't say for sure. Now we will check over the line data equals pi divided by two to check this symmetry. We will let both our R and our data be transformed into negative R and negative data. And we will check this substitution. You will get negative R equals six minus six sine negative data. By our odd function, we get negative R equals six plus six sine theta. And by dividing both sides by negative one to get R by itself, we get R equals negative six minus six sine data. That is not the same as our original function. So similarly, we cannot say that it has symmetry over the line beta equals pi over two. Finally, we will check Paul Symmetry for Paul Symmetry. We see the C if we replace our R with negative R, do we get the same answer back out? So we have NR equals six minus six sine theta divide both sides by negative one. We get R equals negative six plus sine theta, which is not the same as our original. So we cannot confirm if it has Pole symmetry. Now that we've checked all of our symmetries, we can graph the power equation slush. Just find some test values to put on a graph data an hour in a table and we will find some angles to use or a graph. Let's say we use zero pilo ban by six pilo ban by three P band by two, two pi divided by 35 P divided by six and pie. And we'll also use some more values going to Tupi being seven pi divided by six or pi divided by three, three pi divided by two five pi divided by three 11 pi divided by six. And finally two pie that we would just plug all these values into the art equation. We use the calculator which I've done here. We get 0. 0.836. And for our bottom rows, we have 9 11.2 12, 11.2 nine and six, we can find those by just throwing into our calculator course. Now we can go ahead and graph our function. We have a point at angle zero, radius six. So we will draw this in the zero degree direction at radius six and we'll do the same for every angle in radius as you go around 563 pi divided by 30.8 P one by two pi divided by 30.8 by pi divided by pi six. Then our next parts seven pi divided by four pi by three, 11.2 three pi divided by two five pi value by 3 11.2 Love and pie divided by six and nine. And finally two pie six, we can now draw a curve connecting all of our points which we notice we get the shape of a cardioid. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Graphing Polar Equations Involving Sine

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