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

Question

In Exercises 27–32, select the representations that do not change the location of the given point.(7, 140°)(−7, 320°)

Accepted Answer

Welcome back. Everyone in this problem. We want to figure out which of the following polar coordinates represents the same location as four and 80 degrees for our coordinates. The first is 4, 440 degrees. The second negative 4, 260 degrees, third, negative 4, 300 degrees and fourth negative four and negative 100 degrees. For our answer choices A says it's two only B says three and four C says 12 and four and D says one and three. Now, if we're going to find which of the polar coordinates represents the same location, let's try to visualize what's going on to, to help us. OK. So let's draw a polar graph here. OK? Or a sketch of a polar graph rather. And in degrees, we know we're gonna be going from zero degrees all the way to 360 180 that's 270 degrees. And the polar coordinate for 80 degrees would probably be somewhere right here. OK. Now, we are trying to find the polar coordinates with the same location and recall that for a polar coordinates R theta where R represents the radial distance and theta represents the angle, then R theta has the same location as R and theta plus or minus 360 degrees multiplied by N where N is an integer. In other words, in other words, we are basically saying that a polar coordinate will have the same location if we add multiples of 360 degrees to it. And that makes sense because 360 degrees represents a complete revolution on a polar graph. So we'll end up at the same place we started. We also can recall that our polar coordinate is going to have the same location as negative R and theta plus 180 degrees. And that also makes sense because negative R means we would negate our radial distance. But it's still going to have the same absolute value. It's still going to be the value of R. The sign just tells us its location. Likewise, if we add 180 degrees to it, we can imagine that a point would come somewhere here, which basically tells us that it's the same, it makes the same angle counterclockwise with the positive X axis. OK. So if we have that point as well, then we can also make complete revolutions. We can also add 360 degrees to that point to figure out which other ones would have the same location. So we have two conditions here. If we compare the polar coordinates that we have notice that I, the first one fits the first condition because we have our value of R which is four and then 23 and four fit the second condition because they all have a value of negative four, the negative value of R for our polar coordinate. So we can compare our first condition. We can look at our first condition to see if it will give us 440 degrees. And in our second condition, we can first change negate the value of R at 180 degrees and then continuously add or subtract 360 degrees to see if we get any of these coordinates. So let's start with the first condition are our first coordinate. So given our original coordinate four and 80 degrees, let's add 360 degrees to it to see if we get 440 no, 80 degrees plus 360 degrees, 80 plus 360 is going to be the same thing as uh 36 plus eight, which is 44. And then we put back our zero. So that's 440 degrees. So 480 has the same location or represents the same location as four and 440 degrees. Therefore, our first coordinate represents the same location. Let's test the next three. First, let's turn, let's negate our value of R and add our 180 degrees. So we're saying that four and 80 degrees will have the same location as negative four and 80 degrees plus 180 degrees and 80 plus 180 equals 260. So that's negative four and 260 degrees. But already just by doing that, OK, we realize that that is our second coordinate negative 1, 260 degrees. So it also represents the same location. No, we just need to compare it for our third and our fourth coordinates. Now our third coordinate is greater than 260 degrees. So let's add 360. But I think just by looking at it, you should get an idea of what our answer will be. So if we take negative four and 260 degrees, OK? And we're going to add 360 degrees, OK, then we're going to end up with uh 620 degrees. OK? So we'll end up with negative four because 360 plus 260 equals 620. So that's negative 4, 620 degrees. 620 degrees is much larger than 300 degrees. So that means negative 4, 300 would not represent the same location. Finally, let's try for a negative four and a negative 100 degrees since negative 100 is less than 260 that means we can subtract 360. So we're saying that it's going to be equal to negative four and 260 degrees. Minus 360 degrees. No 260 minus 360 equals negative 100. So negative four, negative 100 degrees will have the same location as negative 4, 260. Thus the same as four and 80 degrees. So coordinate four is also a correct choice. Therefore, options 12 and four are correct options and that means our correct answer is 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.(7, 140°)(−7, 320°)

Key Concepts

Polar Coordinates

Angle Measurement

Negative Radius in Polar Coordinates

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