Question

Determine two pairs of polar coordinates for the point (2, -2) with 0° ≤ θ < 360°.

(2 square root of 2, 225°), (-2 square root of 2, 45°)

(2 square root of 2, 135°), (-2 square root of 2, 315°)

(2 square root of 2, 315°), (-2 square root of 2, 135°)

(2 square root of 2, 45°), (-2 square root of 2, 225°)

Answer 1

So let's get our first point then. Now, all the answers already gave you the 2 sqrt2 part, but in case we forgot how to get that, we just have to recall that it comes from the fact that x^2 + y^2 = r^2.
Now to find the angle, we need to be able to relate both x and y to an angle. This can be done with the fact that tan(theta) = y/x So:
\[\tan \theta = \frac{ y }{ x } -> \tan \theta = \frac{ -2 }{ 2 } -> \tan \theta = -1\]
So do you know where tan(theta) = -1?

Answer 2

315 and 135?

Answer 3

@Psymon

Answer 4

Correct. Now since we need two polar coordinates, we need both of those angles. So if our original rectangular coordinates are (2,-2), this would be in the 4th quadrant where the angle would be 315 degrees. So one answer is (2sqrt2, 315).
Now we need a second, unique answer. So what (2rt2, 315) is saying is that when we face in the direction of 315 degrees, we move forward 2sqrt 2 units. But since we already did that, the other option is to face in the opposite direction and walk backwards 2sqrt 2 units. So what 2nd point would represent facing the other direction and walking backwards?

Answer 5

Oh gosh..this is all so confusing. Wouldn't it be with 135?

Answer 6

@Psymon

Answer 7

Well, does the first answer make sense?

Answer 8

mmm, no not really

Answer 9

@Psymon

Answer 10

Okay, we'll backtrack then. So, we know what quadrant (2,-2) is in, correct?

Answer 11

yes @Psymon

Answer 12

Okay, cool. And you mentioned yourself how tan(theta) turns out to be 135 and 315, those are both correct. Now, that (2,-2) is a fixedpoint, it's not going anywhere, it's just the question basically wants you to write it in 2 different ways.

Answer 13

So now it's just realizing what a point in polar coordinates is tellign you. A polar coordinate is (r,theta), where theta is the direction you're facing in and r is basically how much you move forward. So that point, if I face in the direction of 315 degrees, is 2rt2 steps away. That kind of make sense?

Answer 14

Yeah a little bit now.but i'm confused as to if it has a limit of what points can be used, and I have two options. how do i differentiate which one to use? @Psymon

Answer 15

Do you mean to say that if there are two ways to answer, how do you know which way to use?

Answer 16

And I'm here, no need to tag me each time :P

Answer 17

okay sorry haha i'm new to this thing and I thought I'd get better answers than yahoo since no one explains. and yes!

Answer 18

Yeah, this is better than yahoo, haha. And honestly, there is no correct way. Logically, most people would pick positive answers as opposed to negative, but never know. When we get our second answer, hopefully youll understand why one answer makes more sense than the other :3
But yeah, for the first point of (2rt2, 315), that make sense a bit?

Answer 19

Yess that visual helped!

Answer 20

Alright, awesome. So now there are two way in polar coordinates that you can reach a single point. You can either face it and walk forward up to it OR you could turn around and walk backwards!.



Now if you walk backwards, your radius is negative. Think you can see the second point knowing you're looking the other way and going backwards?