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

Solution:

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

We know that,

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

r = √(x² + y²) and θ = tan^-1(y/x)

We have,

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

Therefore,

r = √(3² + 3²) = √(18) = 3√2

and

θ = tan^-1(3/3)

θ = tan^-1(1)

θ = tan^-1(π/4)

θ = tan^-1tan[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°.

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

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

Summary:

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