In Exercises 13–34, test for symmetry and then graph each polar e...

Question

In Exercises 13–34, test for symmetry and then graph each polar equation.

r = 4 sin 3θ

Accepted Answer

The polar equation for symmetry and sketch its graph where R is equal to seven halves sign of three. The and we're also given a polar coordinate system to graph on now to test our polar equation, we need to test three types of symmetry. Polar symmetry pole symmetry and the line theta equals pi divided by two symmetry. Let's first check our polar to test polar. We will replace our data with negative data and see if we get the original equation back out after we do our substitution. So we can replace our data with negative data making this negative three the inside. And now we can make use of a sign identity. In particular, we know sign is an odd function. If we know that we can say sign at a negative angle, it equals to negative sign of the angle but positive, which is a characteristic of odd functions. No, let's go ahead and replace our equation. Since we said this is negative sign, this turns into negative seven halfs sign of positive three theta. This is not the same as our original. So we cannot confirm that we have polar symmetry. Let's go ahead and check our pole symmetry. Next for this, we will replace our radius R with negative R. So we get negative R equals seven halves stein three, the divided by negative one to get rid of the negative gives us R equals negative seven halves sine three theta which once again is not the same as the original. So we cannot confirm pulse symmetry. Let's finally check symmetry over the line data equals pi divided by two. For this, we will replace both RR and our theta. We get negative R equals seven halves sine of negative three. The now based on earlier, we know our negative three, the transforms us to get negative seven halves sine positive three. The and finally, we divide both sides by negative one, we get R equals seven halves sine three data. Now we notice this is the same as our original. So we can confirm we have symmetry over the line theta equals pi divided by two. This means we will have symmetry on our vertical axis here from left to right that we've done that we can go ahead and graph our equation. Now we will need input and output device which I have found a few here. And so we can go ahead and plot these on our graph. Since we have symmetry, we only need to check values from zero to pi normally we would check 0 to pi because the period of sine goes from 0 to 2 pi But in this case, we will just check zero to pi because we know we have symmetry. So for our first one, for example, we would have R equals seven halves sign of three, theta where theta is zero, you get seven halves sine zero, where we know sine zero is zero, this whole thing goes to zero. So we will plot a point at the equals zero and R equals zero. Now with the plot on a polar coordinate system, we would go in the direction of data and we would move in that direction. The number for R for example, since we're in the direction of zero, which is towards the right, our R zero, we would put a point right in the middle. Our next one is pi divided by 63.5 in the direction of pi divided by six, we will move 3.5 is roughly right there. Pi divided by 42.47 would be next and we just keep doing this in order. So we have all of our points. I divide it by 30 pi divided by two negative 3.5 with a negative value. We will go the opposite direction of our angle two pi divided by 30 C pi divided by 42. five pi divided by 63.5. And finally pi zero. Now we notice this looks a little bit weird. We would need to make a smooth curve between all these points to see exactly what it looks like. But we would go in order and we would kind of curve out our points. This kind of looks something like this where we curve out, hit our point curve in and do the same thing for every one of our points giving us a graph that looks something like this. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

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In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ