In Exercises 19–21, find the product of the complex numbers. Leave answers in polar form.z₁ = 3(cos 40°+i sin 40°)z₂ = 5(cos 70°+i sin 70°)

Verified step by step guidance

1. Identify the given complex numbers in polar form: \(z_1 = 3(\cos 40^\circ + i \sin 40^\circ)\) and \(z_2 = 5(\cos 70^\circ + i \sin 70^\circ)\).
2. Recall that the product of two complex numbers in polar form \(z_1 = r_1(\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i \sin \theta_2)\) is given by \(z_1z_2 = r_1r_2(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))\).
3. Calculate the product of the magnitudes: \(r_1r_2 = 3 \times 5\).
4. Add the angles: \(\theta_1 + \theta_2 = 40^\circ + 70^\circ\).
5. Express the product in polar form using the calculated magnitude and angle: \(z_1z_2 = 15(\cos(110^\circ) + i \sin(110^\circ))\).

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 magnitudes are multiplied together, and the angles are added. For example, 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 makes it straightforward to compute products without converting back to rectangular form.

Trigonometric Identities

Trigonometric identities, such as the sine and cosine addition formulas, are essential for simplifying expressions involving angles. For instance, 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 from the multiplication of complex numbers. Understanding these identities is crucial for accurately expressing the final result in polar form.