Graph each polar equation. Also, identify the type of polar gr...

Question

Graph each polar equation. Also, identify the type of polar graph.

r = 2 + 2 cos θ

Accepted Answer

Welcome back, everyone. Graph the polar equation R equals 5 + 5 cosine theta and identify the type of graph it produces. So first of all, we want to notice that this graph has a form of R equals a plus B cosine theta, right? And essentially if the ratio between A and B in its absolute value is exactly 1, which is the case here, right, because if we take the absolute value, our A is 5, our B is 55 divided by 5 is equal to 1, right? If this condition holds, then our graph is considered. To be a cardioid. So this is the type of graph, and now what we're going to do is basically plot it, right, because we want to graph the polar equation. Well, essentially we are introducing a polar plane and we want to evaluate our function R or different values of data. So if we evaluate R at 0. We're going to substitute 0 into the given equation, and this gives us 5 + 5 cosine of 0. And if we evaluate the result, well, essentially this gives us 10 because cosine of 0 is equal to 1, right? So now if we consider our angle of 0, R is going to be equal to 10. We want to add that point. Now we're going to move on to pi divided by 6. We're going to choose our perfect angles. If we evaluate R at pi divided by 6, we're going to get approximately 9.3, right? So we can write down approximately. 9.3, right? So what we want to do is basically consider that angle and add a value of about 9.3. Then we're going to consider pi divided by 3. This is going to be our next angle. It's pie divided by 3. The value of R is exactly 7.5. So what we want to do is locate 7.5. We have our 7, and then we're going to add 7.5, right? Then we're going to consider the angle. I divided by 2, and the value of R is exactly 5. From here, let's consider 2 pi divided by 3. We get 2.5. So we want to add 2.5. Well done, we have our point, followed by 5555 divided by 6. At this angle, the value of R is 0.7, so it's really close to zero, right? We can essentially add that value, very close to the origin. Then we're going to consider i at this point we have 0. And then moving on, because it is a cardioid, we will have a symmetrical shape, right? So what we're going to do is basically connect these points. Let's go ahead and do that. So first of all, we are going to connect the first two points, or by the next one. Right, we, we have the first half and now we symmetrically want to draw the other half. So let's make sure that we draw it more accurately because we have a larger precision closer to the origin. And now let's symmetrically draw the other half, right? So what we're going to do I simply draw the reflection of this shape relative to the horizontal axis. Now, to increase precision, we are just going to fix the first part of the graph, right, just to make sure that we include all of the points that we have. And now from here we to increase precision, we can actually calculate. Additional angles. So let's go ahead and do that just to make sure that we draw it really accurately. Our next angle is going to be 75 divided by 6. At that angle, we get a value of about 0.7, so once again we get the same 0.67. Then we have an angle of 4 divided by 3, which gives us 2.5, so we want to add 2.5. Followed by an angle of 3 by divided by 2. We get 5 Followed by 5 pi divided by 3. We get 7.5. And then we take 11 divided by 6. Where we have approximately 9.3. So now we want you at approximately 9.3. And that's it. Now we are just going to connect those points. Which finishes our graph. Now let's make sure that we connect them more accurately, just to make sure that all of them are connected. And that's it. We have the final graph. Thank you for watching.

Graph each polar equation. Also, identify the type of polar graph.
r = 2 + 2 cos θ

Key Concepts

Polar Coordinates

Polar Equations

Graphing Polar Equations

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