# Determine two pairs of polar coordinates for the point (3, -3) with 0° ≤ θ < 360°?

We will use the relation between Cartesian Coordinates and Polar Coordinates to solve this question.

## Answer: The two pairs of polar coordinates for the point (3, -3) with 0° ≤ θ < 360° are (3√2, 45°) and (3√2, 225°).

Go through the explanation to understand better.

**Explanation:**

We know that,

the relationship between the Cartesian Coordinates (x, y) and the Polar Coordinates (r, θ) is given below:

r = √(x^{2} + y^{2}) and θ = tan^{-1}(y/x)

We have,

The rectangular coordinates (x, y) = (3, 3)

Therefore,

r = √(3^{2} + 3^{2}) = √(18) = 3√2

and

θ = tan^{-1}(3/3)

θ = tan^{-1}(1)

θ = tan^{-1}(π/4)

θ = tan^{-1}tan[nπ + (π/4)]

Here n is an integer and π is the period of tan function.

θ = [nπ + (π/4)] = 180°n + 45°

Now substitute the value of n = (-1, 0, 1, 2, … etc.) and get the angles

If n = -1, then θ = -180° + 45° = -135°

If n = 0, then θ = 0° + 45° = 45°

If n = 1, then θ = 180° + 45° = 225°

If n = 2, then θ = 360° + 45° = 405°, etc.

Since 0° ≤ θ < 360, therefore the value of θ is 45° and 225°.