Stitz-Zeager_College_Algebra_e-book

# Our nal answer is x 2 cost y 3 sint for 2 t 2

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: he following theorem summarizes the advantages of working with complex numbers in polar form. Theorem 11.16. Products, Powers and Quotients Complex Numbers in Polar Form: Suppose z and w are complex numbers with polar forms z = |z |cis(α) and w = |w|cis(β ). Then • Product Rule: zw = |z ||w|cis(α + β ) • Power Rule:a z n = |z |n cis(nθ) for every natural number n • Quotient Rule: a z |z | = cis(α − β ), provided |w| = 0 w |w| This is DeMoivre’s Theorem The proof of Theorem 11.16 requires a healthy mix of deﬁnition, arithmetic and identities. We ﬁrst start with the product rule. zw = [|z |cis(α)] [|w|cis(β )] = |z ||w| [cos(α) + i sin(α)] [cos(β ) + i sin(β )] We now focus on the quantity in brackets on the right hand side of the equation. [cos(α) + i sin(α)] [cos(β ) + i sin(β )] = cos(α) cos(β ) + i cos(α) sin(β ) + i sin(α) cos(β ) + i2 sin(α) sin(β ) = cos(α) cos(β ) + i2 sin(α) sin(β ) + i sin(α) cos(β ) + i cos(α) sin(β ) Rearranging terms = (cos(α) cos...
View Full Document

## This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

Ask a homework question - tutors are online