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

Question

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

r² = 4 cos 2θ

Accepted Answer

Welcome back, everyone. Graph the polar equation R2 equals 9 sin to theta and identify the type of graph it produces. So let's notice that this graph has a form of R2 equals 9 assigned to theta, and we can essentially tell that the general form is R2 equals a 2 because 9 is equal to 32, right multiplied by sin of 2 theta. And we have to recall that whenever we have this form, the graph corresponds to a luminous ca. Right, where A is equal to 3. Now we are going to solve for R and because we have R squared, if we take square root of both sides, well, we're going to get two solutions for R positive and negative. So R is equal to positive or negative square root of 9 sin to data. And now to plot the graph, well we are going to introduce a polar plane and we're going to evaluate. The values of R at different theta values. For example, we can begin with theta of 0, right? And if theta is 0, then R is going to be equal to 0. So we have our first point. Now we're going to consider our perfect angles. We can then continue with pi divided by 6. If we take theta equals pi divided by 6. The value of R is going to be approximately 2.79, 2.8, right? Positive or negative, because we have two solutions. So basically what we want to do. I and Approximately 2.8, right? So pi pi pi divided by 6 is our angle, and now we essentially want to consider 2 and approximately 2.8. So it's close to 3, right? So we're going to add that point and we're going to add that same point in the opposite quadrant. Right, because we have two solutions, positive and negative. Well done. Now we are going to consider the next angle, which is going to be divided by 4. If we take pi divided by 4, the value of R is going to be positive or negative 3. So we want to We want to add a value of positive or negative 3. Let's go ahead and add that value. Now we have exactly 3. Positive or negative way, so we want to also look at the opposite quadrant. Well done. And from here, we are going to consider our next angle, which is pi divided by 2. This is where we get 0, right? Now for values that are 2 divided by 33 pi divided by 4 and 5 pi divided by 6, those are undefined. So, quadrant number 2 is undefined, but for pi we get 0 again. Well by 7 pi divided by 6. Now 7 pi divided by 6. Gives us two solutions, positive or negative. 2.8 approximately, right? So once again we get those two exactly same solutions, right? So we will not get any additional points because we have. A symmetrical function, right? We will essentially have a lemness gate. Now for quadron 4 ones gain, it will have undefined values just as we had in quadron 2. So what we can do is simply connect these points. To obtain the final shape. So we are going to add one. And then the other, and this is how our rough graph looks like. That would be it. Thank you for watching.

Graph each polar equation. Also, identify the type of polar graph.
r² = 4 cos 2θ

Key Concepts

Polar Coordinates

Polar Equations

Graphing Polar Equations

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