embassyhill said:This is supposed to be a proof of trigonometric multiplication of complex numbers:
What happened at the =...= point? I understand everything up to that.
The proof of trigonometric multiplication of complex numbers is based on the use of the polar form of complex numbers, which states that a complex number can be represented as a combination of a magnitude and an angle. This allows for the multiplication of complex numbers to be simplified using trigonometric identities.
The polar form of complex numbers is used in the proof by rewriting the complex numbers in terms of their magnitude and angle. This allows for the use of trigonometric identities, such as the sine and cosine functions, to simplify the multiplication of complex numbers.
Some common trigonometric identities used in the proof of trigonometric multiplication of complex numbers include the double angle identities, product-to-sum identities, and sum-to-product identities. These identities allow for the simplification of complex expressions involving trigonometric functions.
The proof demonstrates the multiplication of complex numbers by showing that the product of two complex numbers can be expressed in terms of the magnitude and angle of the individual numbers. This allows for the use of trigonometric identities to simplify the expression and ultimately arrive at the product of the complex numbers.
Understanding the proof of trigonometric multiplication of complex numbers is important for several reasons. It provides a deeper understanding of the relationship between complex numbers and trigonometry, and it allows for the simplification of complex expressions involving complex numbers. Additionally, this proof is foundational in many areas of mathematics and science, such as engineering and physics.