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 − 3 sin θ

Accepted Answer

Test the polar equation for symmetry and sketch its graph where we have R equals five minus 11 sign of data. And we have a coordinate plane here in polar coordinates. To graph our equation on now to graph this, we need to check three types of symmetry. First, we have polar symmetry P symmetry and symmetry over the line theta equals pi divided by two. Let's first check our pole symmetry. Our polar symmetry, I should say for our polar symmetry, we will replace our data with negative data and see if we get the same equation as our original back out. This gives us R equals five minus 11 sign of negative data. Now we know sine is an odd function. So since sine is an odd function, if we have s sign of a negative angle of data, we can replace that with negative sign of a positive angle, which means in our function here, we would get R equals five plus 11 sine data when taking into account our odd function. Now this is not the same as our original function. So we will say this does not have polar symmetry that we can confirm next let's check our P symmetry for pulse symmetry. We will just replace our R with a negative R. And see if we get the same equation back out. We would have negative R equals five minus 11 sine data divided by negative one ticket R by itself gives us R equals negative five plus 11 sign data, which is also not the same as our original. Finally, we'll replace both R and theta with our negative counterparts. Four symmetry over the line theta equals pi divided by two. We will say negative R is equals to five minus sine negative data. Now we know the sign negative data from before. This would give a SNR equals five plus 11 sine data divided by negative one gives us RS equals to negative five minus 11 sine theta, which also is not the same as our original. So we cannot confirm any of our symmetries since we can't do that. Let's go ahead and move on to the graph. Now to graph this function, we will just need some values to plug into our equation with the, as our independent variable, we will choose values of data. No, we know the period of sign goes to two pipe. So we can choose values between zero and two pi to graph. For instance, we would say R equals five in our equation. It's minus 11 sine theta. So we would say minus 11 sine zero for our first one. That would be five minus 11 sine zero from the unit circle we know is zero. This would give us five. We can do this for the rest of our values which I have put in a table here. We cannot graph all of these on our plane. Our first one is data equals zero or equals five. This means in the direction of angle data, you will plot a point at radius of five or five units in that direction. So our 1st 105, we can put on the breath. Our next one is pi divided by six and negative 0.5. Now pi divided by six, we would go in that direction. But since this is negative, we would go in the opposite direction for the radius, which means we actually had in direction of seven pi divided by six based on our negative radius. We would repeat for every one of our next points. Our next one pi divided by three negative 4. which we can draw a point pi divided by two negative six. To put there two pi divided by three, negative 4.53 55, divided by six, negative 0.5 and pi five that rounds out our first half of our period X we have seven pi divided by 6, 10.5. So we will count up to 10 and going in direction four pi divided by 3, 14.53. Roughly right there three pile valued by 2 16. Just gonna put a point down there 55, divided by 3, 14.53. Love and pi divided by 6, 10.5. And finally two pi five. Now we have to draw a smooth curve between these points. But in order, so we started at 05. So we can draw a curve between these, which we draw a curve connecting all of our points. In order beauty is something that looks similar to this and that is the graph of our equation. 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 − 3 sin θ

Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Graphing Polar Equations Involving Sine

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