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. (2, − 3π/4) (2, − 7π/4)

Accepted Answer

Welcome back everyone. In this problem, we want to figure out which of the following polar coordinates represents the same location as five and negative five fourths of pi first, the first coordinate is five and 3/4 of pi second five and 1/6 of pi third, negative five and negative nine fourths of pi and fourth negative five and a negative two pi for our answer choices A says it's two and four B says it's four only C one and three and D says it's three only. Now, if we're going to find the polar coordinates with the same location, let's at least try to sketch a polar graph to understand exactly what we're looking for. So let's say that this is a polar graph. OK. Let me put that a bit better here. OK. And we know that on our polar graph, it goes from 0 to 2 pi hm. And let's say that the 0.5 and negative five fourths of pi is somewhere right here. Now, what we're really looking for is that we want to find a coordinate that has the same location. OK. Somewhere that it would represent the same location as five and negative 5 4% of pi. So what do we know about that? Well, recall that a polar coordinate that is a coordinate in the form R theta. OK has the same location as a polar coordinate, which is R plus two pi the sorry two pi multiplied by N where N is an integer and that would be plus or minus rather. So what are we seeing here? We're basically saying that the polar coordinate has the same location as another polar coordinate that has added a multiple of two pi. And that makes sense because two pi represents a complete revolution around our polar graph. So we are saying that if we make a revolution, we'll have the same location. OK. We also know we also know that or if we make our polar coordinate, OK, if we make the radial distance of our polar coordinate negative and add pi to our polar coordinate, then that point will also have the same location. In other words, if we go py aliens and in the opposite direction here, let's put that in blue. So let's say that we had a point here, then it would also have this represent the same location as the original polar coordinate. And again, that too makes sense because if we make the radial distance negative, it still has the same absolute value. So it still represents the same distance. And when we add pi to our angle, it still represents the same angle or the same distance counterclockwise from the positive x axis. OK. So if we take these two points and add two pi to the both of them, then we will get an infinite number of points that will represent the same location. Now, if we look at our answer choices here, the first tool have the same value of R as our polar coordinate where R equals five and theta equals negative five fourths of pi. If we look at three and four, notice that they are the negative value of our value are negative five. So if we can use our, what we know in our first scenario that we can add multiples of two pi to check if these two coordinates represent the same location as our theta. And we can use what we know in the second scenario that is negate our add pi and then find add multiples of two pi to it as well to figure out if three and four represent the same location as R theta. So let's go ahead and do that. Let's start with our first scenario where we can add multiples of two pi. Now notice that both of our answers are greater than negative five fourths of pi. So it would make sense to add two pi in this scenario. So given, let me put that in red, OK, given or coordinate five and negative five worths of pie. That means it's going to represent the same location as five and negative five worths of pi plus two pi A plus two P. Now, two pi is the same thing as eight fourths of pi. And we did that because now we have both of them in the same denominator and negative five fourths of pi plus eight fourths of pi is equal to 3/4 of pi. So that means five and 3/4 of pi represent the same location as five and negative five fourths of pi. So one is a correct option. OK. Now how about our second polar coordinate five and 1/6 of pi? Well, we've added one multiple of two pi. OK. And now we've gone from negative five fourths of pi to 3/4 of pi. If we think about that on a number line, what we're seeing is that here from negative five fourth of pi, of course, we know we have zero, the 3/4 of pie is one revolution. When we think about 1/6 of pi, 1/6 of pi would appear before we make that complete revolution. 1/6 of pi would be somewhere right here. OK. So that tells us then that 1/6 of pi could not represent the same location. So 1/6 of pie is not included. Now, since we've looked at our first scenario, let's think about the next one to do that. Let's negate our value of R and add pi to our anger. So we're taking our polar coordinate which is five and negative five worths of pi. And we're saying it will have the same location as negative five and negative five worth soft pie plus pie. Eh No pi is the same thing as four fourths of pi. If we express it with the same denominator and negative five plus four equals negative one. So our coordinate would be negative five and negative worth of pi. So this also represents the same location. So now for three and four, we're going to want to compare negative five and negative fourth of pi to the both of them notice that both of them are less than negative fourth of pi. So we can subtract two pi to do our comparison. So let's take our coordinate negative five negative fourth of pi. And now we're saying that it's going to be the same thing as negative five and negative fourth of pi negative four of pie minus two P. Now we're subtracting two pi. That's the same thing as subtracting eight fourths of pi and negative pi minus eight fourths of pi will be equal to negative nine fourths of pi. So negative nine fourths of pi sorry, negative five and negative nine fourths of pi. That coordinate has the same location as five and negative five fourths of pi. If we look back on our list, that means three is also a polar coordinate that represents the location. Finally let's check for negative five and negative two pi. Now when we think about it, OK, negative nine fourths of pi. Let's put that on a number line here again. So we've gone from negative a negative 1/4 of pi to negative nine fourths of pi. And of course, we have a zero here, right? But now let's think about it. Where would negative two pi fall on this number line? Well, here it would be between negative fourth of pi and negative nine fourths of pi because negative two pi is the same thing as negative eight fourths of pi. OK. So that means negative five and negative two pi would not represent the same location because it comes before we make our first complete revolution. Therefore, the fourth coordinate is not the same location. The only coordinates with the same location are one and three. If you look back on our answer choices, that's answer choice. C 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. (2, − 3π/4) (2, − 7π/4)

Key Concepts

Polar Coordinates and Point Representation

Angle Coterminality in Polar Coordinates

Effect of Negative Radius and Angle Adjustments

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