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°)

Verified step by step guidance

1. Identify the given complex numbers in polar form: \( z_1 = 6(\cos 20^\circ + i \sin 20^\circ) \) and \( z_2 = 5(\cos 50^\circ + i \sin 50^\circ) \).
2. Recall the formula for multiplying two complex numbers in polar form: \( z_1 \cdot z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \).
3. Multiply the magnitudes of the complex numbers: \( r_1 = 6 \) and \( r_2 = 5 \), so \( r_1 r_2 = 6 \times 5 \).
4. Add the angles of the complex numbers: \( \theta_1 = 20^\circ \) and \( \theta_2 = 50^\circ \), so \( \theta_1 + \theta_2 = 20^\circ + 50^\circ \).
5. Express the product in polar form using the results from the previous steps: \( z_1 \cdot z_2 = 30(\cos(70^\circ) + i \sin(70^\circ)) \).

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 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 calculations by allowing the magnitudes to be multiplied and the angles to be added.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, the product is obtained by multiplying their magnitudes and adding their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then the product z₁z₂ = r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂). This property greatly simplifies the process of finding products of complex numbers.

Trigonometric Identities

Trigonometric identities, such as the sine and cosine addition formulas, are essential for simplifying expressions involving angles. For example, the formulas cos(α + β) = cos α cos β - sin α sin β and sin(α + β) = sin α cos β + cos α sin β help in calculating the resultant angle when adding angles in the multiplication of complex numbers. Understanding these identities is crucial for accurately expressing the final result in polar form.