Question

Find the polar coordinates of the point whose cartesian coordinates are `(-sqrt(2), -sqrt(2))`.

Answer 1

Here, `x = -sqrt(2)` and `y = -sqrt(2)`

∴ The point lies in the third quadrant.

Let the polar coordinates be (r, θ).

Then r² = x² + y²

= `(-sqrt(2))^2 + (-sqrt(2))^2`

= 2 + 2

= 4

∴ r = 2

`cos θ = x/r`

= `(-sqrt(2))/2`

= `-1/sqrt(2)`

And `sin θ = y/r`

= `(-sqrt(2))/2`

= `-1/sqrt(2)`

∴ tan θ = 1

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

tan θ = 1 = tan `π/4`

= `tan (π + π/4)`   ... [∵ tan (π + θ) = tan θ]

= `tan  (5π)/4`

∴ `θ = (5π)/4`

Here, the polar coordinates of the given point are `(2, (5π)/4)`.