In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.z₁ = cos 80° + i sin 80°z₂ = cos 200° + i sin 200°

Verified step by step guidance

1. Identify that the complex numbers are given in polar form: \( z_1 = \cos 80^\circ + i \sin 80^\circ \) and \( z_2 = \cos 200^\circ + i \sin 200^\circ \).
2. Recall that the quotient of two complex numbers in polar form \( \frac{z_1}{z_2} \) is given by \( \frac{r_1}{r_2} \text{cis}(\theta_1 - \theta_2) \), where \( \text{cis}(\theta) = \cos \theta + i \sin \theta \).
3. Since both \( z_1 \) and \( z_2 \) have a magnitude of 1 (as they are on the unit circle), the magnitude of the quotient \( \frac{z_1}{z_2} \) is also 1.
4. Calculate the difference in angles: \( \theta_1 - \theta_2 = 80^\circ - 200^\circ = -120^\circ \).
5. Convert the angle \(-120^\circ\) to a positive angle between \(0^\circ\) and \(360^\circ\) by adding \(360^\circ\), resulting in \(240^\circ\). Therefore, the quotient in polar form is \( \text{cis}(240^\circ) \).

Key Concepts

Complex Numbers in Polar Form

Complex numbers can be represented in polar form as z = r(cos θ + i sin θ), where r is the modulus (magnitude) and θ is the argument (angle). This representation simplifies multiplication and division of complex numbers, as it allows us to work with magnitudes and angles directly.

Division of Complex Numbers

To divide two complex numbers in polar form, z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), the quotient is given by z₁/z₂ = (r₁/r₂)(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂). This means you divide the magnitudes and subtract the angles, which is essential for finding the result in polar form.

Argument of a Complex Number

The argument of a complex number is the angle θ formed with the positive real axis, typically measured in degrees or radians. When expressing the argument, it is important to ensure it lies within a specified range, such as between 0° and 360°, to maintain consistency and clarity in representation.