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Polar coordinates of & point are given Flnd other polar coordinates (r, 0) of the polnt that satisfy the following(a)0, -2I < 0 < 0(r, 0) =r < 0, 0 <...

Question

Polar coordinates of & point are given Flnd other polar coordinates (r, 0) of the polnt that satisfy the following(a)0, -2I < 0 < 0(r, 0) =r < 0, 0 < 0 < 2(r, 0) =0, 27 $ 0 < 43(c, 0)

polar coordinates of & point are given Flnd other polar coordinates (r, 0) of the polnt that satisfy the following (a) 0, -2I < 0 < 0 (r, 0) = r < 0, 0 < 0 < 2 (r, 0) = 0, 27 $ 0 < 43 (c, 0)



Answers

Polar coordinates of a point are given. Find other polar coordinates $(r, \theta)$ of the point for which: (a) $r>0,-2 \pi \leq \theta<0$ (b) $r<0,0 \leq \theta<2 \pi$ (c) $r>0,2 \pi \leq \theta<4 \pi$ $$ (-2,3 \pi) $$

The .3 pi over four in 4 other ways. Okay so first let's draw it. Okay three by over four. So on the third circle at 45 degrees. So we want to write it again and we want to keep data positive I mean are positive so our will still be three but we want to give a negative answer name. Okay so let's just call it -75 or four. Okay so three -75 or four. Okay next we want to keep our to be positive so are still three and we want to give it a name Fade up bigger than two pi. Okay we'll just go around once to buy plus pi That's nine pi over four. Okay now we want to give it a name with a negative are but with a positive angle so we got a give this red piece a name. An angle name that's positive. And then when we call up -3 we'll have to go back that way. So his name is 12345 paper for. Okay so watch me do that. I need to go to five pi over 4123455 or four and then go backwards. 3123. Okay so it's the same point. Alright last one we wanted to have a negative name and the angle to have a negative name. Oh that's easy. Now that we did it this way -3 pi over four So go to -3 pi over four. There it is. Go back to -3. There it is

Okay, we're given the polar coordinate one, comma pi over two and then we're asked to find different representations of that coordinate um with different criteria. So first of all, let's just go ahead and plot this. If we go pi over two, that will put us at our top of our unit circle there and then we're just going one unit out. Okay, Now, for part A were asked for a radius that is greater than one. So that has to be our value or greater than zero, so that has to be our value of one and then a negative angle. So instead of going a positive pi over two, we're going to be going the other direction. And so that's going to be a negative three Pi over two. For part B will be using a radius that is less than zero, so we're using our negative one and then our angle has to be positive. So we really have to be across from pi over two. So that's going to be three pi over two. And then when we go backwards will be going back towards pi over two and then our part C has a radius that is greater than zero, so that will be one. And then our angle has to be between two pi and four pi. So we'll go ahead and add two pi to our pi over two angle with a common denominator of two. We can make that a four pi over two. So then when we add them together, we see that we are at five pi over two.

Q. We are given the polar coordinate negative three comma four pi. And we're gonna applaud it and then we're gonna also um answer parts A. B and C. To come up with different representations of the same point. Okay, so for pie, that would be going around the unit circle twice. And so from there instead of going towards zero, um which is equivalent to four pi on our unit circle. Um we're instead going backwards so we're actually going towards pi three units. So for part A it's asking me for first radius that is greater than zero. So that would have to be three. And then it's asking me for a negative angle between um negative two pi and zero. So the quickest way to get there is to go clockwise and go pi over. And so that would be considered a negative pi. So our next part um part B. We have to have a radius that is smaller than zero. So that will have to be negative three. Now the next thing is we need to have an angle between zero and two pi. So remember when you're a negative radius you're actually using an angle on the other side. So the angle straight across would be zero to pi four pi. So notice they've included zero but not too pie. So we'll have to use the value zero now for part C. Were again a positive radius. And so we'll go ahead and put that value in and then we need to be a value between two pi and four pi. So if you go all the way around the unit circle and then consider how much more you have to go. You need to go an extra two pi so that's going to get us to three pi.

Okay, we have a given polar coordinate. And then um we are going to find different representations of this polar coordinate based on criteria in parts A B N C. So let's first of all go ahead and plot to calm a pie. So we're going over pie and then going out to units. So for part A we still need to have a positive radius, so we're still going to be using that value to but then we have to have a negative angle somewhere between negative two pi and zero. So that negative angle will actually be a negative pi. Now, for part B, the radius has to be negative. So let's go ahead and throw a negative two in there. So if we are negative then that means we have to actually face towards the angle on the other side which would be either zero or two pi. But notice our statement include zero but not too pie. So we'll be using zero. And then for part C, um we still have a positive radius so we'll use the value to, and then our angle has to be between two pi and four pi. So if we take our old angle pi and add two pi to it, we'll get ourselves to three pi.


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