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

Question

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

Accepted Answer

Test the polar equation for symmetry and sketch its graph where we have our cosine theta equals negative seven. And we have a coordinate plane to put our equation on. Now to graph this, we need to test our three symmetries. First, this is our polar symmetry, our pole symmetry and our symmetry over the line theta equals pi divided by two. Now our first step will be to check polar symmetry for this. We will replace data with negative data and see if we get the same equation as our original back out. So here we have R cosigned data equals negative seven. Replacing our data gives us our cosine negative data equals negative seven. Now we know cosine is an even function which tells us if we have cosine a sum function X with a negative inside. This is the same thing as saying cosine of X but positive inside we can then instead write our equation as R cosine data equals negative seven, which is the same as our original. So we confirm we have polar symmetry. Now let's check our pole symmetry for pulse symmetry, we will turn our radius or R negative and see what we get out, we have negative R cosine theta equals negative seven divided by negative one to get rid of the negative gives us R cosine theta equals seven. That is not the same as our original. So we can't confirm Pole symmetry. Finally, we will check over the line theta equals pi divided by two. For this, we will change both our R and our data to be negative. We will get negative R cosine negative theta equals negative seven by art even functional, our data will turn positive and we can divide by negative one to get R cosine data by itself, we get our cosine theta equals seven. That is also not the same as the original. So we cannot confirm symmetry over the line theta equals pi divided by two. Now that we know our symmetry, let's go ahead and graph our function. We know symmetrical over the polar axis which is very similar to the X axis and rectangular. So let's go ahead and find some points to graph on. Now, here I have some values on the table for angles of data. So we can just find some values and plug them into our equation. An artist, we have theta and R. Now our equation is actually our cosine theta equals N seven. If we solve for R, we would get R equals negative seven divided by cosine data which would then give us a way to find input and output values. Our first input value is data equals zero. We would have R equals negative seven divided by the cosine of zero. Now we know the cosine of zero is positive one. You, we get negative seven divided by one which is equals to negative seven. We'd repeat that process for different angles. And here I chose the interval from zero to pi since that is a good interval to check for our function. But we see here our first value is at zero negative seven. Now to graph this, we will go in the direction of angle zero and we go to the left by seven. Since we have a negative radius or we'll go away from that angle. So zero is directly to the right. So we'll go to the left by seven. Our next one pi divided by six, negative 8.08. So in our direction, the pi divided by six, we go the opposite direction to negative 8.08. And we will keep doing this for all of our points pi divided by three, negative pi divided by two undefined. So we can't graph that one, two pi divided by three positive 14 five pi divided by 68. MP seven. Now we noticed this is just a vertical line between all the points. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

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

Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Conversion Between Polar and Cartesian Coordinates

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