Lesson 6.5 Trigonometric Form of Complex Numbers

Complex Numbers Recall complex numbers can be written in the form: Also, they are graphed on the “real – imaginary” plane: Real Axis Imaginary Axis Absolute Value of a Complex Distance from (0,0) to (a, b)

Trigonometric Form Remember changing vector components using trig: We can do the same with complex numbers (where r represents the magnitude) Where r is the distance (absolute value), so…

Example Write the complex number in trigonometric form.

Example Write the standard form (a + bi) of the complex number Problem Set 6.5.1

Product & Quotient of Complex Numbers Start with 2 complex numbers: Then the product is: And the quotient is: r = magnitude We could factor out these coefficients

Example Find the product of the complex numbers.

Try It: Take a complex number Multiply it by itself Now, multiply this product by the original See a pattern? Square itCube it

De Moivre’s Theorem For a complex number where n is a positive integer: Remember where the n ‘s go

Example Use De Moivre’s to find: Problem Set 6.5.2