In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.z₁ = 5 (cos 4π/3 + i sin 4π/3)z₂ = 10 (cos π/3 + i sin π/3)

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1. Identify the given complex numbers in polar form: \( z_1 = 5 (\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}) \) and \( z_2 = 10 (\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \).
2. Recall the formula for dividing two complex numbers in polar form: \( \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \), where \( r_1 \) and \( r_2 \) are the magnitudes, and \( \theta_1 \) and \( \theta_2 \) are the angles.
3. Calculate the magnitude of the quotient: \( \frac{r_1}{r_2} = \frac{5}{10} = \frac{1}{2} \).
4. Determine the angle of the quotient by subtracting the angles: \( \theta_1 - \theta_2 = \frac{4\pi}{3} - \frac{\pi}{3} \).
5. Express the quotient in polar form: \( \frac{z_1}{z_2} = \frac{1}{2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \).

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude and angle, represented as z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. This form is particularly useful for multiplication and division of complex numbers, as it simplifies the operations by allowing the magnitudes and angles to be handled separately.

Division of Complex Numbers in Polar Form

To divide two complex numbers in polar form, you divide their magnitudes and subtract their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then the quotient z₁/z₂ is given by (r₁/r₂)(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)). This method streamlines the division process and maintains the polar representation.

Trigonometric Identities

Trigonometric identities, such as the sine and cosine addition formulas, are essential for simplifying expressions involving angles. In the context of complex numbers, these identities help in converting the results of operations back into a standard form, ensuring that the final answer is expressed correctly in polar coordinates, which is crucial for clarity and accuracy.