In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form. z₁ = 6(cos 20° + i sin 20°) z₂ = 5(cos 50° + i sin 50°)

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Recall that when multiplying two complex numbers in polar form, the magnitudes multiply and the angles add. This is based on De Moivre's Theorem.

Identify the magnitudes and angles of the given complex numbers: For \(z_1 = 6(\cos 20^\circ + i \sin 20^\circ)\), the magnitude is 6 and the angle is \(20^\circ\). For \(z_2 = 5(\cos 50^\circ + i \sin 50^\circ)\), the magnitude is 5 and the angle is \(50^\circ\).

Multiply the magnitudes: \(r = 6 \times 5\).

Add the angles: \(\theta = 20^\circ + 50^\circ\).

Write the product in polar form as \(r(\cos \theta + i \sin \theta)\) using the values found in steps 3 and 4.

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Polar Form of Complex Numbers

A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude (modulus) and θ is the argument (angle). This form is useful for multiplication and division because it separates the magnitude and angle components.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, multiply their magnitudes and add their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].

De Moivre's Theorem

De Moivre's theorem states that raising a complex number in polar form to a power n results in rⁿ [cos(nθ) + i sin(nθ)]. While not directly required here, it underpins the manipulation of angles and magnitudes in polar form and helps understand multiplication as repeated addition of angles.