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Multiplication and division of complex numbers in trigonometric form


Some operations with complex numbers are most easily performed when the numbers are in trigonometric form.

Here's how division multiplication operations work.

Multiplication

Consider two complex numbers in trigonometric form. . The product It is given by:

Remembering the arcing formulas:

Like this:

Note that the product is a complex number whose modulus is the product of the factor modules and whose argument is the sum of the factor arguments.

Example
Calculate the product , with :

Resolution

  • The module of it's product .
  • The argument of is given by the sum .

Like this:

 

Division

Consider two complex numbers in trigonometric form:

.

The quotient It is given by:

Remembering the arc difference formulas:

And of the fundamental trigonometric relationship:

Like this:

Note that the quotient is a complex number whose modulus is the quotient of the dividend and divisor modules, and whose argument is the difference of the dividend and divisor arguments.

Example

Calculate the quotient , with :

Resolution

  • The module of is the quotient .
  • The argument of is given by the difference

How We do:
Like this:

Next: Potentiation and Rooting in the Trigonometric Form