In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form.z₁ = cos π/4 + i sin π/4z₂ = cos π/3 + i sin π/3

Verified step by step guidance

1. Step 1: Recognize that the given complex numbers are in polar form, where z₁ = cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}) and z₂ = cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}).
2. Step 2: Use the formula for multiplying complex numbers in polar form: if z₁ = \text{cis}(\theta_1) and z₂ = \text{cis}(\theta_2), then z₁z₂ = \text{cis}(\theta_1 + \theta_2).
3. Step 3: Identify the angles \theta_1 = \frac{\pi}{4} and \theta_2 = \frac{\pi}{3}.
4. Step 4: Add the angles: \theta_1 + \theta_2 = \frac{\pi}{4} + \frac{\pi}{3}.
5. Step 5: Express the product in polar form as \text{cis}(\theta_1 + \theta_2).

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ), represented as r(cos θ + i sin θ). This form is particularly useful for multiplication and division of complex numbers, as it simplifies calculations by allowing the use of trigonometric identities.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, the magnitudes are multiplied, and the angles are added. 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 simplifies the process of finding the product.

Trigonometric Values

Understanding the trigonometric values of common angles is essential for converting between polar and rectangular forms of complex numbers. For example, cos(π/4) = sin(π/4) = √2/2 and cos(π/3) = 1/2, sin(π/3) = √3/2. These values are crucial for accurately calculating the resulting polar form after performing operations on complex numbers.