In Exercises 7–12, test for symmetry with respect to a. the pol...

Question

In Exercises 7–12, test for symmetry with respect to   a. the polar axis.   b. the line θ=π2.   c. the pole.r = 4 + 3 cos θ

Accepted Answer

Hello. Today we're going to be looking at the given co polar equation and determining whether that equation has symmetry with the polar axis. The line theta equal pi over two or the pole of the polar axis. So we are given R is equal to 10 plus seven multiplied by cosine of the. Let's start by checking to see whether the equation has any symmetry with the polar axis. Now, one thing to note is that the polar axis is defined as the x axis of the polar axis. So in order to check whether the equation has symmetry with the polar axis, what we'll need to do is we'll need to replace data with negative data. And if we do so, and we end up with R is equal to 10 plus seven multiplied by cosine data that will prove that we have symmetry with the polar axis. So if we were to plug in negative data for the, the new equation we end up with is R is equal to 10 plus seven multiplied by cosine of negative data. Now recall that the rule of science for cosine states that cosine of negative data is equal to cosine of the. So this trigonometric identity will allow us to eliminate the negative sign from inside the cosine function. And what we are left with is R is equal to 10 plus seven multiplied by cosine of data. Notice that when we made the substitution of negative data for the, we ended up with the same equation that we were given originally that shows that we have symmetry with the polar axis. Next, we're going to check the same condition and see whether we have symmetry with the line theta is equal to pi over two. Now, in order to check whether their symmetry with data is equal to pi over two, we're going to want to replace our comma theta with negative R common negative data or in other words, we're going to replace our R term and the term with negative versions of those terms. And if we end up with the original equation that will show that we have symmetry with the line theta is equal to pi over two. So once we use these substitutions, what we are left with is negative R is equal to 10 plus seven multiplied by cosine of negative data. Now, just like before the law signs for cosine states that cosine of negative data is equal to cosine of data. So we can replace cosine negative data with cosine of data. And that will leave us with negative R is equal to 10 plus seven multiplied by cosine of data. Next, we want to solve for R. And in order to do that, we'll have to divide the entire equation by negative one. So that will leave us with R is equal to negative minus seven multiply by cosine of the. This is the simplified form of the equation and this form of the equation does not match the original equation that proves that there is no symmetry with the line theta is equal to pi over two. Now, what we wanna do is you want to test to see whether we have symmetry with the pole of the polar axis. In order to show that there is symmetry with the pole of the polar axis, we need to replace R with negative R. So if we apply the substitution to our original equation, what we are left with is negative R is equal to 10 plus seven multiplied by cosine of the. Now, just like before we want to sol for R. And in order to do that, we can divide everything by negative one that will leave us with positive R is equal to negative 10 minus seven multiplied by cosine of theta. And just like before, since this equation does not match the original equation that shows that we have no symmetry with the pole of the polar axis. So just to summarize every way that we want to check symmetry, we need to do a specific substitution in order to check to see whether there's symmetry with the polar axis, we need to replace data with negative data. If we want to test symmetry for data is equal to pi over two, we must replace both R and data with its negative versions. And if we want to test symmetry with the pole of the polar axis, we must replace R with negative R. And after doing that, we show that there is only symmetry with the polar axis. So since there is only symmetry with the polar axis, that shows that the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 7–12, test for symmetry with respect to a. the polar axis. b. the line θ=π2. c. the pole.r = 4 + 3 cos θ

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