Express 3 − 4i in polar form z = r(cos θ + i sin θ). What are r and θ (principal value)?

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Multiple Choice

Express 3 − 4i in polar form z = r(cos θ + i sin θ). What are r and θ (principal value)?

To write a complex number in polar form r(cos θ + i sin θ), you need its modulus r and its argument θ. The modulus is r = sqrt(a^2 + b^2) for z = a + bi, and the principal argument θ is chosen in (-π, π]. For z = 3 − 4i, the modulus is r = sqrt(3^2 + (−4)^2) = sqrt(9 + 16) = 5. The tangent of the angle satisfies tan θ = b/a = (−4)/3. Since a > 0 and b < 0, the point is in the fourth quadrant, so θ must be a negative angle. The principal value is θ = arctan(−4/3) = −arctan(4/3). Therefore, the polar form has r = 5 and θ = −arctan(4/3). This places the point in the correct quadrant with the correct magnitude.

To write a complex number in polar form r(cos θ + i sin θ), you need its modulus r and its argument θ. The modulus is r = sqrt(a^2 + b^2) for z = a + bi, and the principal argument θ is chosen in (-π, π].

For z = 3 − 4i, the modulus is r = sqrt(3^2 + (−4)^2) = sqrt(9 + 16) = 5.

The tangent of the angle satisfies tan θ = b/a = (−4)/3. Since a > 0 and b < 0, the point is in the fourth quadrant, so θ must be a negative angle. The principal value is θ = arctan(−4/3) = −arctan(4/3).

Therefore, the polar form has r = 5 and θ = −arctan(4/3). This places the point in the correct quadrant with the correct magnitude.

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