Topic: Complex, Parametric, and Polar Forms
Subtopic: Powers and Roots in Trig Form
Overview
From an intermediate algebra course we know how to expand and simplify a Complex number in the form (a+bi)n via multiple FOILs or using Pascal's Triangle. In this lesson we convert the Complex number to trig form and use a formula to evaluate (r·cisθ)n which is a great time saver!
We also learn something we never learned to do in intermediate algebra, that is to take roots of Complex numbers in trig form nth-root((r·cisθ)=(r·cisθ)1/n. Finding powers and roots of Complex numbers in trigonometric form use one of two formulas called DeMoivre's Theorems.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 9.3.1 Algebraically find the power of a Complex number in trigonometric form using DeMoivre's Theorem of Powers,
zn=(r·cis(θ))n=rncis(nθ) - 9.3.2 Algebraically find the roots of a Complex number in trigonometric form using DeMoivre's Theorem of Roots,
the n nth-roots of z are z1/n=(r1/ncis((θ+2πk)/n)) where k=0,1,2,...,n-1 - 9.3.3 Sketch the roots of a Complex number in trigonometric form on the Complex plane
- 9.3.4 Use trigonometric form and DeMoivre's Theorem of Roots to solve equations that have Complex solutions
Terminology
Define: DeMoivre's Theorem of Powers, DeMoivre's Theorem of Roots