Determine whether the tests for symmetry detect symmetry...

Determine whether the tests for symmetry detect symmetry with respect to (a) The polar axis. Replace (r, ?) by (r, -?). (b) The line ? = ?/2. Replace (r, ?) by (r, ? - ?). (c) The pole. Replace (r, ?) by (-r, ?). Otherwise, indicate that the test is inconclusive. r² = 25 cos 2? Part 1 of 3 (a) The test (Choose one) Part 2 of 3 (b) The test (Choose one)

Transcript

00:01 And we're given the polar equation, r squared equals 25 cosine of 2 theta.

00:05 And we want to check for symmetry.

00:06 So let's begin by checking for symmetry with the polar axis with respect to the polar axis.

00:14 So we can see here that we need to replace r theta with r negative theta.

00:18 So in other words, just replace theta with negative theta.

00:21 So let's go ahead and rewrite our equation here.

00:26 R squared equals 25 cosine of 2 negative.

00:36 Theta, which is going to be 25 cosine of negative 2 theta.

00:46 And for this to simplify further here, when you have to remember that cosine is an even function.

00:52 So that means then that cosine of negative theta is the same as the cosine of theta.

01:03 So therefore, the cosine of negative 2 theta is going to be the same thing as the cosine of 2 theta.

01:12 And since this is the same as our original function, we do have symmetry with respect to the polar axis.

01:20 Let's continue on with the next one.

01:22 We want to check symmetry with respect to the line theta equals pi over two.

01:29 So again, here we need to replace r theta with r pi minus theta.

01:34 In other words, replace the theta with pi minus theta.

01:36 So let's do that.

01:38 R squared equals 25 cosine of, 2 pi minus theta.

01:56 So this is going to be 25 cosine of 2 pi minus 2 theta.

02:05 Now to simplify this, we're going to use this equation right here, our cosine angle difference formula.

02:11 So this is going to be 25 cosine...

Question

Please give Ace some feedback