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

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

• A. r=5; θ≈2.498 rad; z = 5 (cos θ + i sin θ)
• B. r=5; θ≈0.6435 rad; z = 5 (cos θ + i sin θ)
• C. r=4; θ≈2.498 rad; z = 4 (cos θ + i sin θ)
• D. r=5; θ≈7π/6; z = 5 (cos θ + i sin θ)

Answer: (A)

To convert a complex number to polar coordinates, find its modulus r and its argument θ, so that z = r(cos θ + i sin θ) with r ≥ 0 and θ in [0, 2π). For z = -4 + 3i, the modulus is r = sqrt((-4)^2 + 3^2) = sqrt(16 + 9) = 5. The point (-4, 3) lies in quadrant II, so the angle is not the arctan(3/(-4)) directly. Instead, take the reference angle arctan(|3/(-4)|) = arctan(3/4) ≈ 0.6435 rad, and place it in quadrant II: θ = π − 0.6435 ≈ 2.498 rad. Thus z = 5(cos θ + i sin θ) with θ ≈ 2.498 rad. This matches the option with r = 5 and θ ≈ 2.498 rad. The other options differ: one uses the wrong quadrant angle (placing the point in quadrant I), another has the wrong radius, and another uses an angle in quadrant III.

To convert a complex number to polar coordinates, find its modulus r and its argument θ, so that z = r(cos θ + i sin θ) with r ≥ 0 and θ in [0, 2π).

For z = -4 + 3i, the modulus is r = sqrt((-4)^2 + 3^2) = sqrt(16 + 9) = 5. The point (-4, 3) lies in quadrant II, so the angle is not the arctan(3/(-4)) directly. Instead, take the reference angle arctan(|3/(-4)|) = arctan(3/4) ≈ 0.6435 rad, and place it in quadrant II: θ = π − 0.6435 ≈ 2.498 rad.

Thus z = 5(cos θ + i sin θ) with θ ≈ 2.498 rad. This matches the option with r = 5 and θ ≈ 2.498 rad.

The other options differ: one uses the wrong quadrant angle (placing the point in quadrant I), another has the wrong radius, and another uses an angle in quadrant III.