In Exercises 35–44, test for symmetry and then graph each pola...

Question

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

Accepted Answer

Test the polar equation for symmetry and sketches graph where we have R equals seven plus five sine two data. And we have a polar coordinate system to graph it on. Now to check symmetry, we need to do three tests. We're testing our Polar Paul and line of theta equals pi divided by two symmetries. So these are either lines or points that we will be reflecting over with the poll reflecting over the origin point. In some way. Let's go ahead and use our substitution for polar symmetry. We are gonna substitute data with negative data. And our goal is to see if we get the original function back out. If we don't get it back out, we cannot confirm that symmetry. So we will get seven plus five sign of negative two data. If we replace our data, now we know sin is an odd function. This tells us if we have some of some negative value, it's the same thing as saying negative sign of that value but positive. So we can then say this transforms it seven minus five sign of two data. This is not the same as our original function. So we cannot confirm this symmetry. Next, we can check our pole symmetry to check Paul symmetry. We need to see what happens with our radius. If we substitute it to be negative, we have negative R equals seven plus five sine to data, solving for R gives us R equals negative seven minus five sine to data. That is also not the same as our original so that it's not have symmetry right there. Finally, we can check our line beta equals pi divided by two. And this one, we replace both our R and our data. So we will get negative R equals seven plus five sign of negative two data, negative R. And by our first one, we know this is seven minus five sine two data. Finally dividing by negative one gives us R equals negative seven plus five assigned to data. This is also not the same. So we cannot confirm this symmetry. Not that we've done all of our tests. We can find some value supply on our coordinate system. Now, I've created a list of values from 0 to 2 pi and we can go ahead and plot all these on our graph. Our first one as theta equals zero and radius equals seven. This means in the direction of angle data, we all move that many spots towards that angle which would be our radius. In other words, in the direction of zero degrees which is to the right, we will go seven on the radius, we'll do the same for every one of these points at this table. If pi divided by six, 11.3 pi divided by three, same number pi divided by 27, two pi divided by 32.7 five. Pi divided by 62.7 and pi seven. If we keep going, we get seven pi divided by six and 11. four pi valued by three same value di di divided by 27 five pi divided by 32. 11 pi divided by 62.7 and two pi seven. We can now connect all these points with a smooth curve. So this roughly starts around, there goes up and keeps following all the points around. So we get this shape which kind of looks like a peanut. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Trigonometric Functions in Polar Equations

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