In Exercises 27–32, select the representations that do not cha...

Question

In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)

Accepted Answer

Hello, today we're going to be identifying which of the following polar coordinates represents the same location as the given polar coordinate. So what we are given is negative 15 comma five pi and we have four coordinates to check in total, we have negative 15 comma eight pi, we have negative 15 comma negative pi, we have 15 comma eight pi and we have 15 comma four pi. Let's start by checking the first coordinate negative comma eight pi. Now one thing to note is that the angle theta in the given coordinate has a greater value than the original coordinate given to us. So if we wanted to check to see whether these two coordinates are equivalent or not, the first thing we can do is add the value of two pi to the given coordinate. So we are given fif negative 15 comma five pi. And what we're going to do is we're going to add two pi to the given angle theta doing this increases the angle to seven pi. So we have negative 15 comma seven pi. Now the value of A pi is greater than seven pi. So if we wanted to check one more time, we would have to add another increment of two pi to our angle theta. So we're going to take negative 15 comma seven pi and add another increment of two pi to the angle. And that is going to give us negative 15 comma nine pi. One thing to note is that the angle eight pi lies between the angles of seven pi and nine pi. And what this means is that angle one is not an equivalent angle to negative 15 comma five pi. Now let's check to see if the second angle is equivalent. So we're going to be checking negative 15 comma negative pi. So one thing to note is that negative pi is less than the value of five pi. So what we'll need to do is we'll need to take our given angle or given coordinate which is negative 15 comma five pi. And we're going to subtract two pi from this value five pi minus two pi will give us the value of three pi. So we have negative 15 comma three pi, we're going to repeat this process again and see whether we get negative pi or not. So we're going to take negative 15 comma three pi and subtract two pi from the angle theta and that will give us negative comma pi. And if we repeat this process one more time, we have negative 15 comma pi minus two pi which will give us the value of negative 15 comma negative pi. So notice that as we continue to decrease our angle by two pi, we ended up getting the angle negative 15 comma negative pi. What this means is that the coordinate negative 15 comma negative pi is equivalent to our original coordinate. We're going to check the third coordinate which is 15 comma eight pi. Now one thing to note is that the radius for the coordinate is positive compared to the negative coordinate that is given to us. So what we're going to need to do is we're going to have to take the positive equivalent of our original coordinate in order to take the positive equivalent, we can change negative 15 to positive 15. But in order to do this, we're going to have it at, at an angle theta or an angle pi to the given theta value. So what this means is that the equivalent positive version to the given coordinate is going to be 15 comma six pi. So just like before we're going to add two pi to theta and see if that gives us the value of a pi. If we were to add two pie to theta, we get 15 comma six pie plus two pie and six pi plus two pi will give us eight pi. So we end up with 15 comma eight pi which matches the third coordinate. So that tells us that the third coordinate is equivalent to the original coordinate, negative 15 comma five P. Now we're gonna check the final coordinate and the final coordinate is given to us as 15 comma four pi. We know that the positive equivalent version of the original coordinate is 15 comma six pi. However, the theta value for pi is less than six pi. So what we're going to need to do is we're going to have to subtract two pie from the angle six pie, six pi minus two pi will give us four pi. So we have 15 comma four pi. And since this coordinate matches the fourth coordinate that shows that the fourth coordinate is equivalent to our original coordinate. So by checking all of the coordinates, the only given coordinate that does not match the original is negative comma a pi and that means the 2nd 3rd and 4th coordinate is equivalent to our original coordinate which is negative 15 comma five pi. And with that being said, 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.

In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)

Key Concepts

Polar Coordinates and Their Representation

Equivalent Polar Coordinates

Effect of Changing Sign of Radius and Angle

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