Question

Which polar coordinates represent the same point as the rectangular coordinates (-4,5)?

Answer 1

2nd option

Explanation:

To convert from (x, y ) → (r, θ )

where

r =
`√(x^2+y^2)`

θ =
`tan^(-1)` (
`(y)/(x)` )

Here (x, y ) = (- 4, 5 ), a point in the 2nd quadrant

r =
`√((-4)^2+5^2)` =
`√(16+25)` =
`√(41)`

θ =
`tan^(-1)` (
`(5)/(4)` ) ≈ 51.3° ← related acute angle , then

θ = 180° - 51.3° = 128.7°

Thus

(- 4, 5 ) → (
`√(41)`, 128.7° ) in polar form