Question

Determine two pairs of polar coordinates for the point (4, -4) with 0° ≤ θ < 360°.

Answer 1

A polar coordinate is that which can be written as (r, θ) where r is the radius and θ is the angle. 

The radius, r, is also the hypotenuse of the right triangle that can be formed. Hence, it can be calculated through the equation,
    
                   r² = x² + y²
If we are to simplify this for the r alone, we have,
                     r = sqrt (x² + y²)

Substituting the known values,
               r = sqrt ((4)² + (-4)²) = 4√2

The x and y can be related through the trigonometric function, tangent. 
      tan θ = y/x
To solve for θ
   θ = tan⁻¹(y/x) = tan⁻¹(-4/4) = -45° = 315°

Hence, the polar coordinate is (4√2, 315°)