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

1 Answer
bp
Jan 21, 2017

(5, pi-tan^(-1) (4/3) )(5,π−tan−1(43))

(-5, -tan^(-1) (4/3) )(−5,−tan−1(43))

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√32+(−4)2=5. Thus r =5. If alphaα is the reference angle which this radial line OP makes with the x- axis, the tan alpha= 4/3tanα=43. Thus alpha= tan^(-1) (4/3)α=tan−1(43). Now the angle which line OP makes with positive axis would be pi - tan^(-1) (4/3)π−tan−1(43) . Thus (-3,4) is represented by (5, pi-tan^(-1) (4/3)(5,π−tan−1(43))

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)2π−tan−1(43) or, -tan^(-1) (4/3)−tan−1(43) with positive x axis. The polar coordinate (-5, -tan^(-1) (4/3))(−5,−tan−1(43)) also represents the same point (-3,4)

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