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# FP2 Polar equations watch

1. So I have to find the polar equation from the following Cartesian equation : (x-1)^2+(y-1)^2=2 and I did this easily using the substitution of the results x=rcosΘ and y=rsinΘ to obtain the correct solution of r=2(cosΘ+sinΘ). But I also had to state
-1/4 pi <Θ< 3/4 pi and this is my question. On these sort of questions how do I find this interval of validity??? I have no idea about this so I full explanation answer would be useful if possible rather than helping me as I fully do not understand how to get this.
Also how do I find the polar equation of (1/x) + (1/y) = (1/a) where a x and y are all greater than 0. I am getting a=(rcosΘsinΘ)/(sinΘ+cosΘ) but the answer is r=a(secΘ+cosecΘ)
2. (Original post by Mathematicus65)
So I have to find the polar equation from the following Cartesian equation : (x-1)^2+(y-1)^2=2 and I did this easily using the substitution of the results x=rcosΘ and y=rsinΘ to obtain the correct solution of r=2(cosΘ+sinΘ). But I also had to state
-1/4 pi <Θ< 3/4 pi and this is my question. On these sort of questions how do I find this interval of validity??? I have no idea about this so I full explanation answer would be useful if possible rather than helping me as I fully do not understand how to get this.
Also how do I find the polar equation of (1/x) + (1/y) = (1/a) where a x and y are all greater than 0. I am getting a=(rcosΘsinΘ)/(sinΘ+cosΘ) but the answer is r=a(secΘ+cosecΘ)
firstly there are some conventions when you are looking at polars

some define the polar plane in the θ "direction" as 0<= θ <2π
some define the polar plane in the θ "direction" as -π< θ <=π, as a complex argument

"a curve defined in an interval ..."
you do this by inspection and experience.
the interval they give there is the minimum interval in θ which traces one revolution of the circle but you can have a larger θ interval(all you do is you trace the curve over and over) or indeed a different interval

3. (Original post by TeeEm)
firstly there are some conventions when you are looking at polars

some define the polar plane in the θ "direction" as 0<= θ <2π
some define the polar plane in the θ "direction" as -π< θ <=π, as a complex argument

"a curve defined in an interval ..."
you do this by inspection and experience.
the interval they give there is the minimum interval in θ which traces one revolution of the circle but you can have a larger θ interval(all you do is you trace the curve over and over) or indeed a different interval

So there is no particular way of identifying the interval other than through inspection?
4. (Original post by Mathematicus65)
So there is no particular way of identifying the interval other than through inspection?
Not really (mathematics is all about using you brain)
It might appear difficult at first but with practice it is ok.
You are looking for what angles the radial vector traces the curve.
The interval by the way is not unique
5. Okay thank you. So if I take say the angle Θ=pi which is outside of the range in the answer then what happens? Ie. Why is this not valid for the equation I determined?
6. (Original post by Mathematicus65)
Okay thank you. So if I take say the angle Θ=pi which is outside of the range in the answer then what happens? Ie. Why is this not valid for the equation I determined?
it is valid

it just produces the same point as that of Θ=0, which is in the interval they quote.

the interval is not unique.

The Cartesian system is unique but the polar system is not, so the choice of interval is up to the conventions of a book, a board syllabus, a University course etc
7. (Original post by Mathematicus65)
So I have to find the polar equation from the following Cartesian equation : (x-1)^2+(y-1)^2=2 and I did this easily using the substitution of the results x=rcosΘ and y=rsinΘ to obtain the correct solution of r=2(cosΘ+sinΘ). But I also had to state
-1/4 pi <Θ< 3/4 pi and this is my question. On these sort of questions how do I find this interval of validity??? I have no idea about this so I full explanation answer would be useful if possible rather than helping me as I fully do not understand how to get this.
(i) Realise that the function can also be written as 2(root 2) Sin(theta+pi/4)
(ii) Presuming that this is an edexcel FP2 questions, then you will only work with r>0 (or equal to zero .... I really must learn Latex..).
(iii) Considering (i) in conjunction with a standard sin wave (which is greater than zero between 0 and pi, with 0 and pi being zeroes) and then note that due to the translation of pi/4 in (i) then you will set your domain as -pi/4 to 3pi/4 (simple translation)
(iv) also note that this is the only domain required to give all possible unique values. (although as mentioned previously this domain is not unique and you could add n * 2pi, where n is an integer to create an equally suitable domain)

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