Match each equation with its polar graph from choices A–D.
...

Question

Match each equation with its polar graph from choices A–D.

r = cos 3θ

Accepted Answer

Welcome back, everyone. Graph the given polar equation R equals 2 cosine theta. We can recognize that we want to plot a rose graph, right, because it essentially has a form of R equals a multiplied by a cosine of B theta. We want to plot R equals 2 cosine 4 theta, A equals 2, and essentially B equals 4. We can sell the number of petals by taking B and multiplying that by 2, so 2 B equals 8 because B is 4, so we expect to have 4 petals. So what we want to do is basically introduce a coordinate plane, and we want to add multiples of pi divided by 12. We start with 0 followed by pi divided by 12, followed by 2 pi divided by 12, which is pi divided by 6, and so on. So for every value we're going to evaluate R. Starting with 0, we have to substitute 0 for data, and this gives us 2 so let's label 2 for now, followed by 1, then we're going to have -1. -2. -1 again. Followed by 12. One -1. -2, 41. 121. -1, 2, -1. Well by 121. And once again we have a -1. -2, -1, followed by 1, right? And at 0, once again we have 2. So basically those are the values that we get when we substitute each angle into the given function. Now how do we plot it? Well, we can begin with the first one. We know that for theta of 0, R is 2, so we identify the value of 2, right, and we are going to label it. Followed by 1 for pi divided by 12, so we are going to label that point followed by -1 for pi divided by 6, right? So we essentially have a negative value now. So what we want to do is just track our line in the opposite direction and we want to add that negative 1. Let's sleep on -1 or by -2. Followed by -1 again. And now once again we have a positive value for 5 pi divided by 12, right? So 5 pi divided by 12 gives us a value of 1. So let's label 1 followed by 2. We can see that pi divided by 2 gives us R of 2, followed by 1475 divided by 12. Let's label that point. Followed by -1 for 2 pi divided by 3, right? So we want to identify. The opposite quadrant, right, so essentially we can do that by. Extending our line. We want to add a point of -1. So let's remove the line. And simply add -1 followed by -2. Well, by -1 again. And then we have a 1 for 11 pi divided by 6, so let's add 1. Followed by tea for pie. Followed by 1413 pi divided by 12. Followed by -1 for 7 pi divided by 6, so once again it's negative, right? And we basically want to add a point. Of R equals -1. And basically we're going to add it. Now let's remove the line. 5 divided by 4 gives us -2, right? So we want to add a point of -2. Now, 4 pi divided by 3 gives us. -1 So let's see where that is, or pi divided by 3 gives us negative ones, we're going to add that point. 17 pi divided by 12 gives us 1. Now, 3 pi divided by 2 gives us 2. Followed by 19 pi divided by 12, which gives us 1. 55 divided by 3 gives us -1. 7 pi divided by 4 gives us -2. 11 pi divided by 6 gives us -1. 23 pi divided by 12 gives us 1. And finally, at 0 or basically 2 pi, we get to a game, right? So now, we expect to have 8 battles and we can just join all of these points. Essentially we are also going to have values at 0 for let's say pi divided by 8, right? So we can also label that point. This is going to be the center point for our curve. And now we want to connect all of those points to form 8 petals. Let's go ahead and do that. So we have our first one. Now we are going to plot our 2nd 1. So let's connect those points. Let's make sure it's clear to see. Now the 3rd 1. Followed by the 4th 1. Than we have The 5th 1 Let's make sure that it looks better. So for the previous one, we can just start with 2 Now let's Introduce our next one. So we have 5 total. Now, we're going to add the 6th 1. The 7th 1. And now finally the 8th 1. So we basically have our final curve. We have a total of 12345678 petals, and this is how the graph is expected to look like. Thank you for watching.

Match each equation with its polar graph from choices A–D.
r = cos 3θ

Key Concepts

Polar Coordinates and Graphs

Rose Curves and Their Properties

Trigonometric Function Behavior in Polar Graphs

Watch next