In Exercises 35–44, test for symmetry and then graph each pola...

Question

In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2

Accepted Answer

S the polar equation for symmetry and sketch its graph where we have R equals four cosine of theta divided by two. And we also have a polar coordinate system here to graph our equation on. Now for symmetry, we need to check three different tests that is our polar symmetry or line equals theta equals pi divided by two symmetry. And our pole symmetry. Let's start with polar symmetry. We have a substitution we can use, let's go ahead and substitute our data first with negative data where we get R equals four cosine negative data divided by two. Now, our goal is to see if, after we do our substitution, do we get the original problem back out? Or the original formula is the same if we do, we confirm that symmetry. Now, here we have four cosine of negative data divided by two. Now, one thing about cosine is that it's actually an even function. This means if we have cosine of negative X, it's the same thing as seeing cosine of positive X. In other words, we can erect this as our equals four cosine theta divided by two. That is the same as our original So we confirm the symmetry. Our next symmetry is the line beta equals pi divided by two. To check this symmetry, we will replace both our radius R and our angle data. You will get negative art equals four negative data divided by two. Now by art even function, this equals negative R equals four cosine data divided by two. And finally divided by negative one to get R equals four cosine theta divided by two but negative. That is not the same as our original. So we cannot confirm the symmetry. Finally, we can check our pulse symmetry by just replacing R with negative R native R equals four cosine data divided by two, which gives us R equals negative four cosine data divided by two which like the last one is not confirmed. So we know we have symmetry over the polar axis now and we can go ahead and now try and graph our function. Now we can find a table of values and plug them into our equation which I have created here. And I chose values between zero and two pi just because the period of our function goes to four pi. So we can check this from 0 to 2 pi first, let's go ahead and plot these points. Our first point is 04. This means in the direction of zero degrees we'll move in that direction four. So our zero degrees direction is headed to the right, we'll move along this line 41234 right there. Our next point is pi divided by 63. five, divided by 63.86. It's roughly right there. Pi divided by 33. I divided by 22. two. Pi divided by five pi divided by 61.04. And finally pi zero, now we can start on the second row seven pi divided by six, negative 1.04. So since we have a negative radius, we will go in the opposite direction of our angle. So we will have this way four pi divided by three negative two, three pi divided by two negative 2. five pi divided by three negative 3. of pi divided by six negative 3. and finally, two pi negative four. Now, since we said we had polar symmetry, this will also reflect vertically. We will start at zero and draw a curve between all of our points and order draw one that loops around and keeps going. Since we have polar symmetry, we will also reflect it vertically. And that should be the graph to our equation. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2

Key Concepts

Polar Coordinates and Polar Equations

Symmetry Tests in Polar Graphs

Graphing r = cos(θ/2)

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