Question

How do you find the pair of polar coordinates to represent the point whose rectangular coordinates are (-3,4)?

Answer 1

`(5, pi-tan^(-1) (4/3) )`

`(-5, -tan^(-1) (4/3) )`

Explanation:

As shown in the graph below, the point (-3,4) lies in the second quadrant. Its radial distance OP would be `sqrt (3^2 +(-4)^2) =5`. Thus r =5. If `alpha` is the reference angle which this radial line OP makes with the x- axis, the `tan alpha= 4/3`. Thus `alpha= tan^(-1) (4/3)`. Now the angle which line OP makes with positive axis would be `pi - tan^(-1) (4/3)` . Thus (-3,4) is represented by `(5, pi-tan^(-1) (4/3)`)

Now extend OP back wards such that length OP' is same as length OP. The radial distance OP' is -5. OP' makes an angle `2pi- tan^(-1) (4/3)` or, `-tan^(-1) (4/3)` with positive x axis. The polar coordinate `(-5, -tan^(-1) (4/3))` also represents the same point (-3,4)