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

Solution:

The polar coordinate can be written as (r, θ) 

Where r is the radius

θ is the angle

The radius here is the hypotenuse of the right triangle formed.

r² = x² + y²

We can write it as

r = √(x² + y²)

By substituting the values

r = √(4² + (-4)²) = 4√2

We know that

tan θ = y/x

Now by solving for θ

θ = tan^-1 (y/x) = tan^-1(-4/4)

As we know that, tan θ is negative in second and fourth quadrant.

In II quadrant, angle θ = 3π/4.

In IV quadrant, angle θ = 7π/4.

Let us consider positive value of r in IV quadrant .

Hence, polar coordinates points will be (4√2, 7π/4).

Let us consider negative value of r in second quadrant.

Hence, polar coordinates points will be (-4√2, 3π/4).

Polar coordinates points of (4,-4) are (4√2, 7π/4) and (-4√2, 3π/4).

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

Summary:

The two pairs of polar coordinates for the point (4, 4) with 0° ≤ θ < 360° are (4√2, 7π/4) and (-4√2, 3π/4).