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, 7π/6) (−2, −5π/6)

Accepted Answer

Welcome back, everyone. In this problem, we want to figure out which of the following polar coordinates represents the same location as negative six and two thirds of pi. The coordinates are one negative six and four thirds of pi, two negative six and 11 3rd of pi 36 and 11 3rd of pi and 46 and negative third of pi for our answer choices A says that it's four only B says three and four C says one and two and D says it's two only. Now if we're going to find those polar coordinates, let's start by drawing a polar graph. So here, so imagine that we have our polar graph. OK? We support our grid right? And we want to find which polar coordinates have the same location as negative six and two thirds of pi. So that point would possibly be here where negative six is the radial distance R and two thirds of pi is the angle. It makes counterclockwise with the positive X axis. OK. So it would be that point would be here negative six and two thirds of pie. All right, let me put the cord in it beside it. Now, what we want to do is to find an equivalent point or an equivalent polar coordinate that represents the same location. And a point in polar coordinates can be represented in infinitely many ways. Recall, OK. Recall that for any polar coordinate R theta, then it's going to have the same location as the point R and theta plus two pi radiance. OK. Which makes sense because what this is really saying is that we make a complete revolution two pi from our angle. And if we make a complete revolution from our point, we know that we will end up at the same place. So that's what this is saying. If we make a complete revolution, we will end up at the same location. And we also know that our theater will also be equal, also be equal to negative R and theta pi radiance. Now, what this is saying is that if we negate our distance, OK, if we negate our distance and add pi to our, our angle, which should probably end up somewhere here, then it's really saying that we're going to get the same location. OK. It's probably somewhere here. And again, that also makes sense because if we negate or, or, or distance R, then it's going to be the same distance, the same radial distance from the origin. Likewise, if we go pi radiance counterclockwise, then the angle it makes relative to the X axis will still remain the same. OK. So what we need to really figure out then is that from this, from this list of points, how can we know which ones will have the same location? Well, we can compare to the information we have notice here, we already know that R is negative six. So based on that, we can use our first observation here. Let's call this, let's call this I, we can use I or we can compare I to 0.1 and 0.2 because both of them have the same value of R while we can compare I I, OK, where we negate R and add pi radiance to 0.3 and four because notice they're not negative six but they're actually six. So let's add two pi two or, or polar coordinate to see if we get one and 21 or two. And let's add pi to our angle to see if we can get three or four. So let's start with the first one. OK. So give them our coordinate, which is negative six and two thirds of pi. What we're saying is that we want to add two pi to our coordinate to see if we get negative 64 3rd of pi or 11 3rd of pi. So let's just add it continuously add or subtract it to see if we will get that location. So in this case, we're going to add it because notice four p, four thirds of pi and 11 3rd of pi are greater than two thirds of pi. So we're gonna have negative six and two thirds of pi plus two pi. OK? No, we're adding, adding two pi to a fraction. So we can write it in terms of thirds. And the two pi in terms of thirds would be six thirds of pi. OK? So that's six thirds of pie and no, six thirds of pi is going to be equal to sorry. Two thirds of pi plus six thirds of pi is eight thirds of pi, no, eight thirds of pi is not in our list. So let's add two pi again to see if we'll get to 11 3rd of pi. So now, oops, let me get rid of that there. So next, we're going to get negative six and eight thirds of pie this time plus two thirds of pi plus two pi, which is six thirds of pi and eight plus six is 14. So our point is negative six and 14 3rd of pi. No, we're, we're past 11 3rd of pi and we still haven't gotten our location. OK? We haven't gotten any of the points. One and two. So we know one and two are not or one and two do not represent the same location. Now, let's move on to our second scenario. OK? That is we're going to negate, we're going to, we're gonna take negative six and two thirds of pi and we're going to negate it. OK? That is we're going to make negative six positive six and we're gonna test if it gets to three by adding a adding Pi Radians until we get to 11 3rd of pi. While we're going to subtract, subtract to see if we get to negative thirds of pi you guys understand. So in other words, from two thirds of pi, we're gonna add until we get to 11 3rd of pi this to see if it gets to 11 3rd of pi rather. And then we're gonna subtract to see if it eventually gets to negative third of pi. So that's what we're gonna do right now. So let's go ahead and do that. No, let's, we're gonna have positive six and now we're gonna have two thirds of Pi plus Pi Radians. If we write that in terms of thirds, that that's three thirds of Pi Radians and two thirds plus three thirds equals five thirds. We're still not there yet. OK? So let's add Pyridium again. OK? So it's gonna be six and let's put R here, it's gonna be six and five thirds of pi plus three thirds of pi again, and five thirds plus three thirds equals eight thirds. So that's six and eight thirds of pi, OK? Still not there yet. So let's see if we can go again. So we're gonna have six and eight thirds of pi again, plus three thirds of pi which equals six and 11 3rd of pi. OK? So now we have six and 11 3rd of pi, which is in our list. In other words, six and 11 3rd of pi represents the same location as negative six and two thirds of pi. So we know that that one is correct. Finally, let's try the fourth one. So let's try to subtract pyres until we get to six and the negative thirds of pi. So subtracting that means we're now going to have six and two thirds of pi minus A minus pi radiance. That's minus three thirds of pi and two pi minus three pi equals negative pi. So we'll have six and negative a third of pi. But again, that's what four is, that's the value of four. So it also means that six and negative thirds of pi has the same location as our polar coordinate. Therefore, our answer is three and four which would be answer choice. B 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, 7π/6) (−2, −5π/6)

Key Concepts

Polar Coordinates and Point Representation

Equivalent Representations in Polar Coordinates

Effect of Angle and Radius Transformations on Point Location

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