Convert rectangular coordinates (-3, 4) to polar coordinates (r, θ) with r ≥ 0 and θ ∈ [0, 2π).

• A. r = 5; θ ≈ 2.2143 rad
• B. r = 5; θ ≈ 0.9273 rad
• C. r = 3; θ ≈ 2.2143 rad
• D. r = 5; θ ≈ π/2

Answer: (D)

The idea is to rewrite the point in polar form using r = sqrt(x^2 + y^2) and θ where x = r cos θ and y = r sin θ, with θ chosen in [0, 2π).

Compute the radius: r = sqrt((-3)^2 + 4^2) = sqrt(9 + 16) = 5.

Locate the angle. The point (-3, 4) lies in quadrant II (x negative, y positive), so use the reference angle α = arctan(|y/x|) = arctan(4/3) ≈ 0.9273 rad. In quadrant II, θ = π − α ≈ 3.1416 − 0.9273 ≈ 2.2143 rad.

Thus the polar coordinates are (r, θ) = (5, 2.2143) with θ in [0, 2π). The angle π/2 would point straight up and would not point toward (-3, 4), and an angle of about 0.9273 rad would place the point in the first quadrant, not quadrant II, so it’s not correct for this position.