Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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