Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III
Sequences and Infinite Series Power Series Vectors and Geometry of Space Vector-Valued Functions
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis
S = contains supplemental resources
Course: Trigonometry
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