Convert the rectangular coordinates to polar coordinates with r > 0 and 0 ≤ θ < 2π. (9, −9)

(r, θ) = ?

(9√2,7π/4)

Thanks!

To convert rectangular coordinates to polar coordinates, we can use the following formulas:

r = √(x^2 + y^2)
θ = arctan(y / x)

Given the rectangular coordinates (9, -9), we can calculate the polar coordinates as follows:

r = √(9^2 + (-9)^2) = √(81 + 81) = √(162) = 9√2

To determine θ, we need to find the arctan value of the ratio of y to x:

θ = arctan(-9 / 9) = arctan(-1) = -π/4

Since we want 0 ≤ θ < 2π, we need to add 2π to the negative angle:
θ = -π/4 + 2π = 7π/4

Therefore, the polar coordinates for (9, -9) are (r, θ) = (9√2, 7π/4).

To convert rectangular coordinates (x, y) to polar coordinates (r, θ), you can use the following formulas:

r = √(x^2 + y^2)
θ = arctan(y/x)

Let's now apply these formulas to convert the given rectangular coordinates (9, -9) to polar coordinates:

1. Calculate the value of r:
r = √(9^2 + (-9)^2) = √(81 + 81) = √162 = 9√2

2. Calculate the value of θ:
θ = arctan((-9)/9) = arctan(-1) = -π/4

However, we want θ in the range 0 ≤ θ < 2π. To adjust the angle to the desired range, we add 2π to the negative angle:

θ = -π/4 + 2π = 7π/4

Therefore, the polar coordinates (r, θ) are:
(r, θ) = (9√2, 7π/4)