Symmetries and Polar Graphs

Identify the symm...

Question

Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Using the symmetries and the table of values, sketch the curve R is equal to 2 minus 2 multiplied by sin theta and the xy plane. OK, so it appears for this particular problem, we're ultimately asked to sketch this curve, and our curve is R is equal to 2 minus 2 multiplied by sin of theta in our XY plane. So, as you can see, we're given a blank graph that we are going to use to plot our sketch. So now that we know what we're ultimately trying to solve for, we need to recall a note that in order to determine the symmetries and to sketch the curve that will be defined by the polar equation R is equal to 2 minus 2 multiplied by sin theta, we will need to first identify what type of curve we are dealing with. And we need to note that based on what we can see, we need to recognize that we are dealing with a cartinoid. And let's write that down. So, our specific graph that we're dealing with is a cardinoid, and this is a specific type of lamacon, which we can write that one down too. Solemicon, and we need to note that this lemicon. Once again, this is a specific type of Lion, and we need to note that the ratio of the constants is going to be 1 since, once again, 2 divided by 2 is equal to 1. So considering these key elements, our first step that we need to take is we need to perform our symmetry analysis. So we will need to test for 3 standard types of symmetry within a polar coordinate system. So we need to consider first the symmetry. So once again we're considering 3 standard types. So we'll call our first type Type A. And for type A, we'll have symmetry that has once again symmetry with respect to the Y axis. So our line that states that theta is equal to pi divided by 2, that means we need to replace parentheses R, theta. With parentheses of R, comma i minus theta. Fantastic. So, that will mean that R is equal to 2 minus 2, so it's gonna be 2 minus 2 multiplied by sin of pi minus theta. And we need to note that sense, since we need to note that sense sign. Of pi minus theta. is equal to sin of theta. That will mean our equation will become r is equal to 2 minus 2 multiplied by sine of theta. So that will mean that our equation will be unchanged, and because our equation will be unchanged, we need to note, therefore that the curve will be symmetric with respect to the y axis. So now, moving right along here to part B now. So we need to consider symmetry with respect to the x-axis, so we need to take R, theta and replace it with R negative theta. So that will give us R is equal to 2 minus 2 multiplied by sin of negative theta, which will be equal to 2 + 2 multiplied by sin of theta. And this is not equivalent to the original equation, and we need to note that when we are replacing with negative, i minus theta, we need to note that this also fails, which will mean as a result, the curve is not symmetric with respect to the x axis. Now considering our part C. Which is symmetry with respect to the pole, which is the origin point. We need to replace R with negative R. So r is equal to 2 minus 2 multiplied by sin of theta, which will mean that R is equal to -2 + 2 multiplied by sin of theta. And once again, this is not equivalent to the original equation, which will mean as a result, the curve is not symmetric with respect to the pole. So moving on to our second step, which is creating a table of values. So when we use the Y-axis symmetry, we can calculate the values from theta is equal to -2. 2, so once again. I should be taking a moment here to pause. So we need to note that we are using our y-axis symmetry, and we can calculate the values from theta equal to pi divided by 2 to theta is equal to positive pi divided by 2. So we're going from pi divided by 2 to positive pi divided by 2, and we will need to reflect them. And when we use these values, we will be able to produce the following chart or table. So I've gone ahead. And created a table for us to refer to, and you can use whatever style of table that you like, but I thought that this style of table made the most sense. And as you can see in our far left column, we have theta and theta will be equal to divided by 2 to divided by 6. 0 pi divided by 6 and positive pi divided by 2 and then we'll have r is equal to 2 minus 2 multiplied by sin of theta and plugging in our values for theta in our expression and going down in order we'll have 4321, and 0, which will yield. The following points in order, 4, pi divided by 23, negative pi divided by 6, 2.0, 1, pi divided by 6, and 0, pi divided by 2. And when we take these points, so let's take our points and we plot them on our graph and we connect them with a smooth curve, we will produce the following graph. So I've gone ahead and used digital graphing software such as Desmos or MATLAB to produce the following graph. And as you can see, we have our points from our table. We have 1, pi divided by 6, we have 2.0, we have 3, negative pi divided by 6, we have 0, pi divided by 2, and we have 4, negative pi divided by 2. And as we can see, we have our curve, and our curve forms a heart shape, and we need to note that this heart shape is formed because of the, and let's make a note off to the side in blue, this is due to the negative sign of theta term, and because of this negative sign of theta term, the The shape is oriented vertically, with the cusp, which is the pointy bit, pointing upward at the pole, and the bulk of the curve resting in the lower half plane, in our 3rd and 4th quadrants. And that's it. That is our final answer. Hooray, we did it. We solved for our final answer. So That's it. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

Symmetries and Polar GraphsIdentify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.r = 1 + 2 sin θ

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Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ