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! :)
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.
In polar coordinates, a point is represented by an angle (θ) and a distance from the origin (r). To convert from rectangular coordinates (x, y) to polar coordinates, you can use the following formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
Let's go through the two examples you provided to understand how to determine the θ values.
1) For the first example, the rectangular coordinates are (-3, 4).
- Calculate r: r = √((-3)^2 + 4^2) = 5
- Calculate θ: θ = arctan(4/-3) = -0.9273 (rounded)
You correctly determined the first θ value as -0.9273. To find the second value, you need to add π (pi) to the original θ value, which yields:
θ2 = -0.9273 + π ≈ 2.2143
Therefore, the two θ values are approximately -0.9273 and 2.2143. Now, to determine which θ value corresponds to which r value, you can compare the signs of the x and y components in the rectangular coordinates.
For the point (-3, 4):
- If x is negative and y is positive, it falls in the second quadrant. So, the first θ value belongs to this point.
- If x is negative and y is negative, it falls in the third quadrant. So, the second θ value belongs to this point.
Hence, the two sets of polar coordinates are (5, -0.9273) and (-5, 2.2143). Notice that the order of r and θ has no significance, as long as they are correctly assigned to the corresponding quadrant.
2) In the second example, the rectangular coordinates are (7/4, 5/2).
- Calculate r: r = √((7/4)^2 + (5/2)^2) ≈ 3.0516
- Calculate θ: θ = arctan(5/2) / (7/4) ≈ 0.9601
Here, you correctly found one of the θ values as approximately 0.9601. However, in this case, you don't need to add 2π (or π) to it since it already falls within the range of 0 ≤ θ < 2π. Therefore, there is no need to find a second θ value.
To determine which r value corresponds to which θ value, you need to consider the sign of the x and y components.
- If x is positive and y is positive, it falls in the first quadrant. So, the positive r value corresponds to this point.
- If x is negative and y is negative, it falls in the third quadrant. So, the negative r value corresponds to this point.
Therefore, the polar coordinates are (-3.0516, 4.1017) and (3.0516, 0.9601).
The reason for adding 2π (or π) to the original θ value is to find all possible angles that correspond to the same point in polar coordinates. In a complete circle, there are infinitely many θ values that can represent the same point. By adding multiples of 2π (or π), you cover all these possibilities.
Remember to always consider the signs of the x and y components to assign the correct quadrant and sign to the r and θ values. I hope this clarifies the process for you!
No problem! I'll be happy to explain.
1) In the first problem, you correctly found the values of r and θ. Since r can be positive or negative, the two sets of polar coordinates are (r, θ) = (5, 2.2143) and (-5, 5.3559). Here's how to know which θ value corresponds to which r value:
- When r is positive, the corresponding θ value is the one you obtained, which is 2.2143.
- When r is negative, you add π (pi) to the obtained θ value. In this case, you add π to -0.9273, which gives you 2.2143.
Therefore, the polar coordinates (5, 2.2143) correspond to the rectangular coordinates (-3, 4), and the polar coordinates (-5, 5.3559) correspond to the rectangular coordinates (-3, 4) as well.
2) In the second problem, you correctly found the values of r and θ. However, you made a small mistake when adding π and 2π. Let me clarify:
- When finding the polar coordinates for the point (7/4, 5/2), you obtained r = ±3.0516 and θ = 0.9601.
- Since r is positive, the obtained θ value, 0.9601, is the corresponding θ value.
- To find the second set of polar coordinates, you add π to the obtained θ value, which gives you 0.9601 + π.
So, the two sets of polar coordinates are (3.0516, 0.9601) and (3.0516, 0.9601 + π). Therefore, the polar coordinates (3.0516, 0.9601) correspond to the rectangular coordinates (7/4, 5/2), and the polar coordinates (3.0516, 3.1017) correspond to the rectangular coordinates (7/4, 5/2).
In summary, when r is positive, you use the obtained θ value as is. When r is negative, you add π to the obtained θ value to get the corresponding θ value. This is because adding π effectively rotates the point by 180 degrees.