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 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.
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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.
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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.
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