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 = 2 − 3 sin θ

Accepted Answer

Test the polar equation for symmetry and sketch its graph where we have R equals four minus seven sine theta. And we also have a polar coordinate system to graph on. Now to test our symmetry, we need to test three different lines or points. We first have polar symmetry and then pole symmetry. And finally symmetry over the line data equals pi divided by two. Now to test polar symmetry, we will make a substitution. We'll substitute our angle data for negative data and we'll see what happens from there. Now we want to see if we get the original function back out after we do our substitution. So let's replace data with negative data. You would get four minus seven sign of negative data. Now we don't want a negative angle here. So we can use one of our identities. In this case, we know sign is an odd function, which means if we have signed of a negative angle, we can erect this as a negative sign of that angle but positive. This means our equation transforms into four plus seven sine theta. Now this is not the same as our original equation, which means we cannot confirm we have symmetry. Now we can try this for a pulse symmetry. For pulse symmetry, we will replace our R with negative R and see if we get the same equation. After solving for R again, we get negative R equals four minus seven sine theta divided by negative one will give us R equals negative four plus seven sine data. That is not the same as the original. So we cannot confirm Pole Symmetry either. Finally, we will check symmetry over the line beta equals pi divided by two. We will replace both our R and our, the with their negative counterparts, we get negative R equals four minus seven sign of negative data. We can use our substitution again giving us four plus seven sine theta. And finally, we divide by negative one to get R equals negative four minus seven sine data, which once again is not the same as the original. Now, we can't confirm any of our symmetries. So now we can just move directly to graphing to do that. We just need some input and output values. We have R equals four minus seven sine theta. Now, since the period of sign goes from 0 to 2 pi we can choose values between zero and two pi or inputs. Let's say we choose zero. First, you would get four minus seven sign of zero what you give us just for by itself. Now, I do have a table here of a bunch of values between zero and two pi we'll plug those in and get out our R values. Now, we can go ahead and graph to graph, we'll plot a point in the direction of our angle and move towards that direction with our value of R. For example, our data is zero and our R is four. That means in the direction of angle zero, we will move four units from the origin 1234, but end up right about there. Now, we would do the same for every one of our points. We have pi divided by six and 0.5. Next pi divided by three negative 2.06. Now, since we have a negative radius, we will actually go in the opposite direction. So start from your origin divided by three with go negative pi divided by two, negative three means we go down two pi divided by three negative two five pi divided by 60.5. And finally pi four gives our first half of values. Now we have seven pi divided by 67.5. So we have moved down and to the left 7. four pi divided by 3, 10. which we would go right about there 11 pi divided by six, 7.5. That's another one. Actually skip one here, actually skip two. Here we want three pi divided by 2 which we count 89, 10, 11 in five pi divided by 10. and finally two pi four, that is all our points which we can now go ahead and connect with a smooth curve. We can start from 04 and a strong curve between all of our points like so, 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 = 2 − 3 sin θ

Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Graphing Polar Equations Involving Sine

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