Question

Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.

Answer 1

Here x = 1 and y = `- sqrt(3)`

∴ The point lies in the fourth quadrant.

Let the polar coordinates be (r, θ).

Then r² = x² + y² = `(1)^2 + (- sqrt(3))^2` = 1 + 3 = 4

∴ r = 2 ...[ ∵ r > 0]

`cos θ = x/r = (1)/(2)`

and `sin θ = y/r = - sqrt(3)/(2)`

∴ tan θ = `- sqrt(3)`

Since, the point lies in the fourth quadrant and 0 ≤ θ < 2π

tan θ = `- sqrt(3) = - tan  π/3`

= `tan(2π - π/3)`  ...[ ∵ tan(2π – θ) = –tan θ]

= `tan  (5π)/3`

∴ θ = `(5π)/3`

∴ The polar coordinates of the given point are `(2, (5π)/3)`.