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

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.

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.