In Exercises 61–63, test for symmetry with respect to a. the po...

Question

In Exercises 61–63, test for symmetry with respect to   a. the polar axis.   b. the line θ = π/2.   c. the pole.r = 5 + 3 cos θ

Accepted Answer

Consider the following equation R equals nine plus five cosine data perform the tests for symmetry with respect to the polar axis. The line data equals pi over two and the pole and we have four possible answers three with symmetry over our respective tests and D with no symmetry. Let's see if we can find the answer to this problem. Now let's first test with respect to the polar exits the test for the polar axis. We will let beta equals to negative data. In other words, we will replace data with negative data and see what happens. We get R equals nine plus five cosine negative data. Now we want to see if this gives us the same as our original equation. Now, we know for a fact that cosine is an even function. If we know that cosine didn't even function, we know cosine negative data, it equals to cosine data. So we can rewrite this as R equals nine plus five plus sine data, which is the same as our original equation. So polo symmetry is checked. Now we will test the line theta equals pi divided by two. The theta equals I divided by two we wanna check if both R and theta are negative, we wanna see what happens to our equation. So let's plug these in. We have R equals nine plus five cosign data which turns into negative R equals nine plus five cosine of negative data. We will get R equals nine minus five cosine negative data with a negative in front of our nine. If we were to move our negative over to the other side. And then by our previous rule for even functions, we have R equals negative nine minus five cosine data. This is not the same as our original function. So we do not know if it is symmetric with the respect to the line, theta equals pi divided by two. So we don't know for sure. It means we will check the pole next to check our pole. We will let RR be substituted with negative R. We get negative R equals nine plus five cosine data and solving or R by itself, we get R equals negative nine minus five cosine data which similarly to our previous check, we don't know for sure if it's symmetrical over the pole. So we can only confirm that we have polar symmetry. In other words, our answer is answer a symmetry with respect to the polar axis only. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

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In Exercises 61–63, test for symmetry with respect to a. the polar axis. b. the line θ = π/2. c. the pole.r = 5 + 3 cos θ

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