# Find the polar co-ordinates of point whose Cartesian co-ordinates are (1,3) - Mathematics and Statistics

Sum

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

#### Solution

(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 1st quadrant.

∴ θ = pi/3

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

Concept: Solutions of Triangle
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