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

Question

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

r = 2 cos θ

Accepted Answer

Test the polar equation for symmetry and sketches graph where we have R equals eight cosine data. And we have a graph here. We can use to plot our function. Now, we need to check our symmetry to check our symmetry. We have to check three different tests. Let's first check our polar symmetry proposed symmetry. We will just replace our data with negative data and see if we get the same function as our original back outs. We have equals eight cosine negative data. Now cosine is an even function because we know this is an even function. We know that cosine of a negative angle just gives us cosine of a positive angle. Other words, they give us the same answers if you are negative and positive angles. In other words, you will say R equals eight cosign data as a result, which is the same as our original because it's the same, we will say this passes polar symmetry. Now when the check symmetry over the line data equals pi divided by two. To check this, we will take both our radius R and our theta and flip the signs and ress substitute them into the equation we will see if we get the same answer back out, we get negative R equals eight cosine negative data which we get negative R equals a cosine data based on our even function. And finally divided by a negative one gives us R equals eight but negative. This is not the same as the original. So we can't confirm if it has symmetry over the line that equals pi divided by two, which means we move on to our final symmetry, which is pole symmetry. For pulse symmetry, we see the check if we change our radius negative. Do we get the same answer back out are the same function. They're the same R equals eight cosine data with negative in front of A R divided by negative one gives us R equals negative eight cosine data. That is not the same as our original. So we can't confirm we have pole symmetry either. So we only confirm that we have polar symmetry. No, we can go ahead and graph our function by finding input and output values plus input value of data to get radius values back up. And we can make a table here with data on top or on the bottom. We just want a few angles to throw on here. We know it has polar symmetry. So we can check angles from zero to pi zero pi divided by six pi divided by three pi divided by 22 pi divided by 35 pi divided by six. And finally, pi now if you plugged in zero, you will get eight cosine zero, which gives us eight. Now we can do the same using a calculator or are you in a circle? But we will get four squares of three, 40, negative four, negative four squares of three and negative eight. Now we can go ahead and plug these points onto our graph. This graph will look like a circle. We plot our points 08 pi divided by 64 squares of three, which is roughly about seven, almost pi divided by pi divided by two pi divided by three negative four five P value by six and negative four squares of three. And finally, pi made of eight connect all of our points and we get a circle. 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 = 2 cos θ

Key Concepts

Polar Coordinates

Symmetry in Polar Graphs

Graphing Polar Equations

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