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. (4, 120°) (−4, 300°)

Accepted Answer

Welcome back. Everyone in this problem. We want to figure out which of the following polar coordinates represents the same location as eight and 210 degrees. The coordinates are first eight and 390 degrees second, negative eight and 50 degrees. Third eight and 410 degrees and fourth negative eight and 30 degrees. For our answer choices. A says it's four only B one only C two and three and D one and four. Now, if we want to find a polar coordinate that represents the same location, let's try to draw a polar graph to get a sense of what we're really looking for. OK. So let's just sketch one over here. OK? And now we know that we're working in degrees. So we're going from 0 to 360 degrees. OK? And no 8, 210 degrees. Sorry, that should be 270 down here. 8, 210 degrees is probably going to be somewhere right here. OK. Let's put that in red. Now, what do we know about location and polar coordinates? Well, recall, OK, recall recall that for a polar coordinate in the form our theater, it will have the same location as the coordinate R and theta plus or minus a multiple of 360 degrees. In other words, if we add or subtract a multiple of 360 degrees, we'll end up with the same location. And that makes sense because 360 degrees represents a complete revolution on a polar graph. Again, we also know that for the point R theta, it has another representation which basically says to get the same location, we can negate our add 180 degrees to theater and then continue to add multiples of 360 degrees to that location. And that also makes sense because R represents the radial distance. In this case, R is eight. And if we had, if we negate R sorry, it will still have the same radial distance. The same absolute value negative just tells us where the coordinate is. If we add 180 degrees, then it's still going to have the same, the angle is still going to be the same size counterclockwise from the positive X axis. And once we do that, then we can make a rotation of two pi and start to end up in this and also end up in the same place as that coordinate. OK. Now if we compare our four coordinates, notice that the first one and the third one match the condition for our theater or, or, or first condition here where we add a multiple of 360 degrees. On the other hand, the 2nd and 4th coordinate match the criteria for our second condition where we negate our notice, both of them are negative eight and we add 180 degrees. So we can test each of these coordinates. We can check by adding multiples of 360 degrees to see if they will give us the same location as eight and 210 degrees. Now, let's start with our first condition. The one in red here where we add or subtract 360. Notice that for both of them, they're greater than 360 degrees. So, oh sorry, they're greater than 210 degrees rather. So let's add 360 degrees. So now that means eight and 210 degrees, we have the same location as eight and 210 degrees plus 360 degrees. No, 210 plus 360 equals 570. Now, when we think about it, OK, if we were to put that on a number line to go from 210 to 570 degrees, our first complete revolution, we would bypass 390 degrees and we'd also pass 410 degrees. That means neither of them can represent the same location. So we know that one is not in the list and three is not in the list. No. How about our second condition for two and four. Well, first, let's negate our at 100 and 80 degrees and then start to add our complete at our subtract our complete evolutions to see if we end up with the same location. So taking it and 210 degrees, we're saying we're going to sneak it and then add 180 degrees to 210 degrees, which is now going to be equal to negative eight and 390 degrees. Now, if we look at coordinates two and four, notice that both of them are less than 390 degrees. So instead of adding 360 let's subtract to see if it will get to any of those. So when we do that, so we'll have negative 8, 390 degrees and it's going to share the same location as negative eight and 390 degrees minus 360 degrees, which is equal to 30 degrees. So 8 210 degrees will represent the same location as negative 830 degrees. That is coordinate four. However, for coordinate two, again, 50 degrees comes before 30 degrees. So it comes before it makes the complete revolution. It's not exactly that complete revolution. So we know it would be cast out as well. It wouldn't be the correct answer. Therefore, the only poor polar coordinate that represents the same location as 8, 210 degrees is coordinate four A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 27–32, select the representations that do not change the location of the given point. (4, 120°) (−4, 300°)

Key Concepts

Polar Coordinates

Equivalent Polar Coordinates

Effect of Transformations on Point Location

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