Question

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

Answer 1

(x, y) ≡ `(1, sqrt(3))`   .......[Given]

Using x = r cos θ and y = r sin θ, where (r, θ) are the required polar co-ordinates, we get

1 = r cos θ, `sqrt(3)` = r sin θ

Now, r = `sqrt(x^2 + y^2)`

= `sqrt(1 + 3)`

= 2

and tan θ = `("r" sin theta)/("r" cos theta)`

= `sqrt(3)/1`

= `sqrt(3)`

= `tan  pi/3`

∴ θ = `"n"pi + pi/3`, n ∈ Z    .......`[(∵ tan theta = tan alpha  "implies"),(theta = "n"pi + alpha","  "n" ∈ "Z")]`

For polar co-ordinates, 0 ≤ θ < 2π

∴ θ = `pi/3` or θ = `pi + pi/3 = (4pi)/3`

But the given point lies in the 1^st quadrant.

∴ θ = `pi/3`

∴ The required polar co-ordinates are `(2, pi/3)`.