Final Answers:
1. The goal of converting z to polar form is: a. Simplify calculations
2. The product zw is calculated using De Moivre's Theorem through: c. Multiplication
3. De Moivre's Theorem allows us to: b. Simplify trigonometric functions
Explanation:
Converting a complex number to polar form involves expressing it in terms of its magnitude (r) and argument (θ), primarily aimed at simplifying calculations involving multiplication, division, powers, and roots. By representing complex numbers in polar form, especially when dealing with multiplication or division, the operation becomes a straightforward task involving the product of magnitudes and the sum of angles, facilitating quicker computations. This method eases mathematical operations and is particularly useful in electrical engineering, physics, and various scientific disciplines where complex numbers are prevalent.
De Moivre's Theorem, used to compute the product zw, involves the multiplication of two complex numbers represented in polar form. This theorem states that raising a complex number to a power (n) involves multiplying its magnitude raised to the power and the angle multiplied by the power. Therefore, for calculating zw, one multiplies the magnitudes of z and w and adds their angles, adhering to the principles of De Moivre's Theorem.
Furthermore, De Moivre's Theorem allows simplification of trigonometric functions involving complex numbers, enabling the calculation of powers and roots efficiently in trigonometric equations involving complex quantities. This theorem becomes a fundamental tool in handling complex numbers within mathematical and scientific contexts, offering a concise method to manipulate and solve problems involving these entities.