Polar Coordinates and finding points

[kylera]

This is an example problem that I can't understand how the answer came out to be this way:
Q: Sketch the polar curve \Theta = 1.
A: A picture of a line that goes diagonal with points that go (1, 1) (2, 1) (3, 1) etc.

I do understand that if the angle is 1, then the line is such that it's 1 radian above the polar axis. What I do NOT understand is how the points come out to be that way. Are those arbitrary numbers, or can I give random numbers? Supposing there was another question like that had a value of 3 instead of 1 for angle, would the points be (1, 3) (2, 3) etc?

 

[HallsofIvy]

This is an example problem that I can't understand how the answer came out to be this way:
Q: Sketch the polar curve \Theta = 1.
A: A picture of a line that goes diagonal with points that go (1, 1) (2, 1) (3, 1) etc.

I do understand that if the angle is 1, then the line is such that it's 1 radian above the polar axis. What I do NOT understand is how the points come out to be that way. Are those arbitrary numbers, or can I give random numbers? Supposing there was another question like that had a value of 3 instead of 1 for angle, would the points be (1, 3) (2, 3) etc?

Yes, that is correct. Remember what (1, 1), etc. mean in polar coordinates. The pair of numbers corresponding to a point gives the r and \theta values: (r, \theta). Saying that \theta= 1 means that r, the first number in each pair, can be anything at all while \theta, the second number, must be 1. If you had, instead, the equation \theta= 3, that would correspond to pairs (in polar coordinates) of (r, 3) where r can be any number. The first number is "arbitrary" because your equation does not mention r; r can be anything. Similarly, if you had an equation that said r= 1, then you would have a curve containing the points (1, 0), (1, 2), etc. (which would be a circle with center at the origin and radius 1.)

(I will confess that I had started writing "No, that's completely wrong!" because I was automatically thinking that the points (1, 1), (2, 1), etc. were given in Cartesian (x, y) coordinates! Of course, in Cartesian coordinates, (1, 1), (2, 1), etc. would lie on the line y= 1.)

 

[kylera]

Right! Correct me if I'm wrong, this means that given the fixed angle, the 'r' value is any value that places distance between it and the origin, right?

 

[HallsofIvy]

Yes, that is correct.