Question

1) I'm asked to find 2 sets of polar coordinates for the point for 0≤ θ< 2pi from a given pair of rectangular coordinates . For instance, one of the coordinates was (-3,4). I found r = plus and minus 5.Then θ= arctan (y/x)= arctan (4/-3)= -.9273
I know you add pi and 2pi to this value to get the two values of θ. This yields 2.2143 and 5.3559. How do you know which θ value corresponds to which r value? The answer given is (5, 2.2143) and (-5, 5.3559).

2) There was another problem. The given rect. coordinates were (7/4, 5/2). I found r to equal plus and minus 3.0516. θ=arctan (y/x)= arctan (5/2) / (7/4)= .9601
My confusion here was that I added pi and 2pi to .9601 just like I'd done in the problem above, expecting it to give me my two θ values, not knowing that it was already one of my θ values. I was only supposed to add pi to it. The answer given was (-3.0516, 4.1017) and (3.0516, .9601). My question is why was this problem different from the one above? I don't know when you're supposed to add pi and 2pi. I don't know why it's done to begin with actually.

Thanks in advance for any help you may be able give this knucklehead! :)

Answer 1

remember that while rectangular coordinates define one and only point, the position of a point using polar coordinates is not unique.
In general any point (r, Ø) can also be represented by (-r, π+Ø)

e.g. (using degrees)
look at the point defined by (6, 150°)
to get to that point, I would go 6 units in the direction of 150° or I could go (150+180)° or 330° and go in the opposite direction , then 6 units
so (6, 150°) is equivalent to (-6,330°)
in radians that would be (6, 5π/6) <-----> (-6, 11π/6)

for your (-3,4), the radius is 5, you had that
I usually get the angle in standard position by ignoring the negative sign, then using the CAST rule to place the angle in the quadrant matching the given point.
so arctan(4/3) = .9273
but (-3,4) is in II , so my angle = π - .9273 = 2.2143
which gives us the first result of (5 , 2.2143)
and our second point is (-5 , 2.2143+π)
or (-5 , 5.3559)
Of course by adding 2π or multiples of 2π , you are just adding rotations, putting you in the same spot
e.g. (5, 2.214 + 6π) would end up at the same point.

Now to your second point: (7/4 , 5/2)
or (1.75 , 2.5) since we are going to decimals anyway.
the positive r value is 3.0516 (you had that)
angle in standard position = arctan (2.5/1.75) = .9601, and our point is in I
so one answer is (3.0516, .9601)
now add π and make the r negative, to give you
(-3.0516 , .9601+π) or (-3.0516 , 4.1017)
YEAHHH ,

Since the angle is in quadrant I, the answer from your calculator was already one of the needed angles, so you just had to add π and change the sign on your r.

Adding or subtracting multiples of 2π just adds or subtracts rotations, the sign on r would not change if you work with multiples of 2π

let me know if you want me to do another example.