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

Solution:

Polar coordinates can be represented as (r, θ) where r equals the radius and θ equals the angle in degrees or in radians.

To find polar coordinates for the cartesian coordinate (4, -4) we have to find out the value of r.

To find r value, use the formula from pythagoras theorem

r² = x² + y²

Where x is the x- coordinate and y is the y-coordinate.

To find r,

⇒ r² = 4² + (-4)²

⇒ r² = 16 + 16

⇒ r² = 32

⇒ r = √32

⇒ r = 4√2

To find out θ, use the formula known as tangent function.

⇒ tan θ = y/x

To find θ,

⇒ tan θ = -4/(4)

⇒ tan θ = -1

⇒ θ = - 45° or -𝜋/4

Since the degree is not within the given limit i.e.0° ≤ θ < 360°,

we have to change the answer to an equivalent answer by adding 360° to the obtained degree answer.

⇒ θ = -45° + 360°

⇒ θ = 315°

One pair of polar coordinate is (4√2, 315°)

To find another polar coordinate subtract 180° from 315°.

⇒ θ = 315° - 180°

⇒ θ = 135°

Now change r value from 4√2 to -4√2.

Second pair of polar coordinate is (-4√2, 135°)

Therefore, two pairs of polar coordinates are (4√2, 315°) and (-4√2, 135°).

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

Summary:

Two pairs of polar coordinates for the point (4,-4) with 0° ≤ θ < 360° are (4√2, 315°) and (-4√2, 135°).