Convert the following Cartesian coordinates to the cooresponding polar coordinates using positive r.
(i) `(1,-1)`
(ii) `(-3,-4)`

Answer

(i) `(sqrt2,-pi//4)` or `(sqrt2,7pi//4)`
(ii) `(5,pi=tan ^(-1)4//3)` or `(5,-pi+tan^(-1)4//3)`

Explanation

(i) Given cartesian coordinates are `(1,-1)`
For r to be positive ,
`r=sqrt(x^2+y^2)=sqrt(1^2+1^2)=sqrt(2) and tantheta =(y)/(x)=-1 `
Since the point `(1,-1)` lies in the fourth quadrant , we can choose `theta=(pi)/(4)`.
So, polar coordinates are `(sqrt(2),-pi//4)`.
Thus the possible poalr coordinates are `(sqrt(2),-pi//4)`, another possibility is `(sqrt(2),7pi//4)`.
(ii) Given Cartesian coordinates are `(-3-4)`.
for r be positive,
`r=sqrt((-3)^2+(-4)^(2))=5`
and `tan theta =(-4)/(-3)=(4)/(3) or theta=tan^(-1).(4)/(3)`
Since the point `(-4,-3)` lies in the third quadrant, we can choose `theta=pi+tan^(-1).(4)/(3)or theta=-pi+tan^(-1).(4)/(3)`.
Thus,the possible polar coordinates are `(5,pi+tan^(-1).(4)/(3))`, another possibility is `(5,-pi+tan^(-1)(4)/(3))`.