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. (−5, − π/4) (−5, 7π/4)

Accepted Answer

Hello, today we're going to be finding the similar polar coordinates to the given polar coordinate. So what we are given is negative 10 comma negative pi over six. And what we wanna do is you want to identify which of these four polar coordinates are equivalent to negative 10 comma negative pi over six. Notice that the radius of the given polar coordinate is negative and we have two polar coordinates that have a negative radius. So we can start by checking coordinates two and coordinates four. So coordinate two is given to us as negative 10 comma 15 pi over six and coordinate four is given to us as negative 10 comma 11 pi over six. If we want to check to see which of these two coordinates is equivalent to the original one, we can take our original coordinate which is negative 10 comma negative pi over six. And we can add increments of two pi to the angle theta negative pi over six plus two pi will give us the value of pi over six. So the new coordinate we get is negative 10 comma 11 pi over six. Notice that after we added the increment of two pi to the angle theta we got the same coordinate as coordinate four. What this means is that coordinate four will be an equivalent coordinate to negative 10 comma negative pi over six. Now we're going to continue this process by adding another increment of two pi to 11 pi over six. So we get negative 10 comma 11 pi over six plus two pi and 11 pi over six plus two pie will give us the value of 23 pie over six. So we have negative 10 comma 23 pi over six. Notice that when we added another increment of two pi, we never got the value of 15 pi over six. Rather we got a value that is greater than 15 pi over six. What this means is that coordinate two is not equivalent to negative 10 come a negative pi over six. Now, what we're going to do is we're going to need to check coordinates one and coordinate three. But one thing to note is that coordinate one and coordinate three have a positive version of the given radius. So what this means is that we're going to have to take the positive equivalent coordinate to negative 10 comma negative pi over six. In order to do this, what we can do is we can change our radius to positive 10. But in order to do that, we're going to have to add a value of P to the given theta value. So we have 10 comma negative pi over six plus pi and negative pi over six plus pi will give us the value of five pi over six. So the positive equivalent to negative 10 comma negative pi over six will be 10 comma five pi over six. We can use this coordinate to check to see if coordinates one and three are equivalent to the original coordinate. So again, we are checking 10 comma 17 pie over six and we're going to check 10 comma 13 pie over six. Let's go ahead and begin that process. Now, just like before we're going to take the original coordinate which is comma five pi over six. And we're going to add an increment of two pi to the angle theta five pi over six plus two pi will give us the value of pi over six. So we have 10 comma 17 pi over six. Notice that after adding an increment of two pi, we get the same coordinate as coordinate one. What this means is that coordinate one is going to be an equivalent coordinate to negative 10 comma negative pi over six. We're going to add another increment of two pie to 17 pie over six. So we get 10 comma 17 pie over six plus two pie and 17 pi over six plus two. Pi will give us the value of 29 pi over six. So we get the coordinate 10 comma 29 pi over six. And one thing to note is that this value of the is greater than the value of data in coordinate three. What this means is that the coordinate three is not going to be an equivalent coordinate to negative 10 comma negative pi over six. So just to summarize, we've identified that out of the four coordinates, the equivalent coordinates are going to be coordinates one and four. 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. (−5, − π/4) (−5, 7π/4)

Key Concepts

Polar Coordinates and Their Representation

Equivalent Polar Coordinates

Effect of Angle and Radius Transformations on Point Location

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