8.6.4 Roots of Unity Flashcards by Irina Soloshenko | Brainscape

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The nth roots of the complex number z = r (cosθ + i sinθ) are given by the formula for k = 0,1…, (n –1)

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Pre-Calculus (300 decks)

1.1.1 The Top Ten List of Mistakes

1.2.1 Concepts of Inequality

1.2.2 Inequalities and Interval Notation

1.3.1 Properties of Absolute Value

1.3.2 Evaluating Absolute Value Expressions

1.4.1 An Introduction to Exponents

1.4.2 Evaluating Exponential Expressions

1.4.3 Applying the Rules of Exponents

1.4.4 Evaluating Expressions with Negative Exponents

1.5.1 Converting between Decimal and Scientific Notation

1.5.2 Converting Rational Exponents and Radicals

1.6.1 Simplifying Radical Expressions

1.6.2 Simplifying Radical Expressions with Variables

1.6.3 Rationalizing Denominators

1.7.1 Determining Components and Degree

1.7.2 Adding, Subtracting, and Multiplying Polynomials

1.7.3 Multiplying Big Products

1.7.4 Using Special Products

1.8.1 Factoring Using the Greatest Common Factor

1.8.2 Factoring by Grouping

1.8.3 Factoring Trinomials Completely

1.8.4 Factoring Trinomials: The Grouping Method

1.9.1 Factoring Perfect Square Trinomials

1.9.2 Factoring the Difference of Two Squares

1.9.3 Factoring the Sums and Differences of Cubes

1.9.4 Factoring by Any Method

1.10.1 Rational Expressions and Domain

1.10.2 Working with Fractions

1.10.3 Writing Rational Expressions in Lowest Terms

1.11.1 Multiplying and Dividing Rational Expressions

1.11.2 Adding and Subtracting Rational Expressions

1.11.3 Rewriting Complex Fractions

1.12.1 Introducing and Writing Complex Numbers

1.12.2 Rewriting Powers of i

1.12.3 Adding and Subtracting Complex Numbers

1.12.4 Multiplying Complex Numbers

1.12.5 Dividing Complex Numbers

2.1.1 An Introduction to Solving Equations

2.1.2 Solving a Linear Equation

2.1.3 Solving a Linear Equation with Rationals

2.1.4 Solving a Linear Equation That Has Restrictions

2.2.1 An Introduction to Solving Word Problems

2.2.2 Solving for Perimeter

2.2.3 Solving a Linear Geometry Problem

2.2.4 Solving for Consecutive Numbers

2.2.5 Solving to Find the Average

2.3.1 Solving for Constant Velocity

2.3.2 Solving a Problem about Work

2.3.3 Solving a Mixture Problem

2.3.4 Solving an Investment Problem

2.3.5 Solving Business Problems

2.4.1 Solving Quadratics by Factoring

2.4.2 Solving Quadratics by Completing the Square

2.4.3 Completing the Square: Another Example

2.5.1 Proving the Quadratic Formula

2.5.2 Using the Quadratic Formula

2.5.3 Predicting the Type of Solutions Using the Discriminant

2.6.1 Solving for a Squared Variable

2.6.2 Finding Real Number Restrictions

2.6.3 Solving Fancy Quadratics

2.7.1 An Introduction to Word Problems with Quadratics

2.7.2 Solving a Quadratic Geometry Problem

2.7.3 Solving with the Pythagorean Theorem

2.7.4 The Pythagorean Theorem: Another Example

2.8.1 Solving a Motion Problem

2.8.2 Solving a Projectile Problem

2.8.3 Solving Other Problems

2.9.1 Determining Extraneous Roots

2.9.2 Solving an Equation Containing a Radical

2.9.3 Solving an Equation with Two Radicals

2.9.4 Solving an Equation with Rational Exponents

2.10.1 An Introduction to Variation

2.10.2 Direct Proportion

2.10.3 Inverse Proportion

2.11.1 An Introduction to Solving Inequalities

2.11.2 Solving Compound Inequalities

2.11.3 More on Compound Inequalities

2.11.4 Solving Word Problems Involving Inequalities

2.12.1 Solving Quadratic Inequalities

2.12.2 Solving Quadratic Inequalities: Another Example

2.13.1 Solving Rational Inequalities

2.13.2 Solving Rational Inequalities: Another Example

2.13.3 Determining the Domains of Expressions with Radicals

2.14.1 Matching Number Lines with Absolute Values

2.14.2 Solving Absolute Value Equations

2.14.3 Solving Equations with Two Absolute Value Expressions

2.14.4 Solving Absolute Value Inequalities

2.14.5 Solving Absolute Value Inequalities: More Examples

3.1.1 Using the Cartesian System

3.1.2 Thinking Visually

3.2.1 Finding the Distance between Two Points

3.2.2 Finding the Second Endpoint of a Segment

3.3.1 Collinearity and Distance

3.3.2 Triangles

3.4.1 Finding the Center-Radius Form of the Equation of a Circle

3.4.2 Finding the Center and Radius of a Circle

3.4.3 Decoding the Circle Formula

3.4.4 Solving Word Problems Involving Circles

3.5.1 Graphing Equations by Locating Points

3.5.2 Finding the x- and y-Intercepts of an Equation

3.6.1 Functions and the Vertical Line Test

3.6.2 Identifying Functions

3.6.3 Function Notation and Finding Function Values

3.7.1 Determining Intervals Over Which a Function Is Increasing

3.7.2 Evaluating Piecewise-Defined Functions for Given Values

3.7.3 Solving Word Problems Involving Functions

3.8.1 Finding the Domain and Range of a Function

3.8.3 Satisfying the Domain of a Function

3.9.1 An Introduction to Slope

3.9.2 Finding the Slope of a Line Given Two Points

3.9.3 Interpreting Slope from a Graph

3.9.4 Graphing a Line Using Point and Slope

3.10.1 Writing an Equation in Slope-Intercept Form

3.10.2 Writing an Equation Given Two Points

3.10.3 Writing an Equation in Point-Slope Form

3.10.4 Matching a Slope-Intercept Equation with Its Graph

3.10.5 Slope for Parallel and Perpendicular Lines

3.11.1 Constructing Linear Function Models of Data

3.11.2 Linear Cost and Revenue Functions

3.12.1 Graphing Some Important Functions

3.12.2 Graphing Piecewise-Defined Functionsv

3.12.3 Matching Equations with Their Graphs

3.13.1 The Greatest Integer Function

3.13.2 Graphing the Greatest Integer Function

3.14.1 Deconstructing the Graph of a Quadratic Function

3.14.2 Nice-Looking Parabolas

3.14.3 Using Discriminants to Graph Parabolas

3.14.4 Maximum Height in the Real World

3.15.1 Finding the Vertex by Completing the Square

3.15.2 Using the Vertex to Write the Quadratic Equation

3.15.3 Finding the Maximum or Minimum of a Quadratic

3.15.4 Graphing Parabolas

3.16.1 Shifting Curves along Axes

3.16.2 Shifting or Translating Curves along Axes

3.16.3 Stretching a Graph

3.16.4 Graphing Quadratics Using Patterns

3.17.1 Determining Symmetry

3.17.2 Reflections

3.17.3 Reflecting Specific Functions

3.18.1 Using Operations on Functions

3.18.2 Composite Functions

3.18.3 Components of Composite Functions

3.18.4 Finding Functions That Form a Given Composite

3.18.5 Finding the Difference Quotient of a Function

4.1.1 Using Long Division with Polynomials

4.1.2 Long Division: Another Example

4.2.1 Using Synthetic Division with Polynomials

4.2.2 More Synthetic Division

4.3.1 The Remainder Theorem

4.3.2 More on the Remainder Theorem

4.4.1 The Factor Theorem and Its Uses

4.4.2 Factoring a Polynomial Given a Zero

4.5.1 Presenting the Rational Zero Theorem

4.5.2 Considering Possible Solutions

4.6.1 Finding Polynomials Given Zeros, Degree, and One Point

4.6.2 Finding all Zeros and Multiplicities of a Polynomial

4.6.3 Finding the Real Zeros for a Polynomial

4.6.4 Using Descartes' Rule of Signs

4.6.5 Finding the Zeros of a Polynomial from Start to Finish

4.7.1 Matching Graphs to Polynomial Functions

4.7.2 Sketching the Graphs of Basic Polynomial Functions

4.8.1 Understanding Rational Functions

4.8.2 Basic Rational Functions

4.9.1 Vertical Asymptotes

4.9.2 Horizontal Asymptotes

4.9.3 Graphing Rational Functions

4.9.4 Graphing Rational Functions: More Examples

4.9.5 Oblique Asymptotes

4.9.6 Oblique Asymptotes: Another Example

5.1.1 Understanding Inverse Functions

5.1.2 The Horizontal Line Test

5.1.3 Are Two Functions Inverses of Each Other?

5.1.4 Graphing the Inverse

5.2.1 Finding the Inverse of a Function

5.2.2 Finding the Inverse of a Function with Higher Powers

5.3.1 An Introduction to Exponential Functions

5.3.2 Graphing Exponential Functions: Useful Patterns

5.3.3 Graphing Exponential Functions: More Examples

5.4.1 Using Properties of Exponents to Solve Exponential Equations

5.4.2 Finding Present Value and Future Value

5.4.3 Finding an Interest Rate to Match Given Goals

5.5.2 Applying Exponential Functions

5.6.1 An Introduction to Logarithmic Functions

5.6.2 Converting between Exponential and Logarithmic Functions

5.7.1 Finding the Value of a Logarithmic Function

5.7.2 Solving for x in Logarithmic Equations

5.7.3 Graphing Logarithmic Functions

5.7.4 Matching Logarithmic Functions with Their Graphs

5.8.1 Properties of Logarithms

5.8.2 Expanding a Logarithmic Expression Using Properties

5.8.3 Combining Logarithmic Expressions

5.9.1 Evaluating Logarithmic Functions Using a Calculator

5.9.2 Using the Change of Base Formula

5.10.1 The Richter Scale

5.10.2 The Distance Modulus Formula

5.11.1 Solving Exponential Equations

5.11.2 Solving Logarithmic Equations

5.11.3 Solving Equations with Logarithmic Exponents

5.12.1 Compound Interest

5.12.2 Predicting Change

5.13.1 An Introduction to Exponential Growth and Decay

5.13.2 Half-Life

5.13.3 Newton's Law of Cooling

5.13.4 Continuously Compounded Interest

6.1.1 Finding the Quadrant in Which an Angle Lies

6.1.2 Finding Coterminal Angles

6.1.3 Finding the Complement and Supplement of an Angle

6.1.4 Converting between Degrees and Radians

6.1.5 Using the Arc Length Formula

6.2.1 An Introduction to the Trigonometric Functions

6.2.2 Evaluating Trigonometric Functions for an Angle in a Right Triangle

6.2.3 Finding an Angle Given the Value of a Trigonometric Function

6.2.4 Using Trigonometric Functions to Find Unknown Sides of Right Triangles

6.2.5 Finding the Height of a Building

6.3.1 Evaluating Trigonometric Functions for an Angle in the Coordinate Plane

6.3.2 Evaluating Trigonometric Functions Using the Reference Angle

6.3.3 Finding the Value of Trigonometric Functions Given Information about the Values of Other Trigonometric Functions

6.3.4 Trigonometric Functions of Important Angles

6.4.1 An Introduction to the Graphs of Sine and Cosine Functions

6.4.2 Graphing Sine or Cosine Functions with Different Coefficients

6.4.3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine

6.4.4 Solving Word Problems Involving Sine or Cosine Functions

6.5.1 Graphing Sine and Cosine Functions with Phase Shifts

6.5.2 Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift

6.6.1 Graphing the Tangent, Secant, Cosecant, and Cotangent Functions

6.6.2 Fancy Graphing: Tangent, Secant, Cosecant, and Cotangent

6.6.3 Identifying a Trigonometric Function from its Graph

6.7.1 An Introduction to Inverse Trigonometric Functions

6.7.2 Evaluating Inverse Trigonometric Functions

6.7.3 Solving an Equation Involving an Inverse Trigonometric Function

6.7.4 Evaluating the Composition of a Trigonometric Function and Its Inverse

6.7.5 Applying Trigonometric Functions: Is He Speeding?

7.1.1 Fundamental Trigonometric Identities

7.1.2 Finding All Function Values

7.2.1 Simplifying a Trigonometric Expression Using Trigonometric Identities

7.2.2 Simplifying Trigonometric Expressions Involving Fractions

7.2.3 Simplifying Products of Binomials Involving Trigonometric Functions

7.2.4 Factoring Trigonometric Expressions

7.2.5 Determining Whether a Trigonometric Function Is Odd, Even, or Neither

7.3.1 Proving an Identity

7.3.2 Proving an Identity: Other Examples

7.4.1 Solving Trigonometric Equations

7.4.2 Solving Trigonometric Equations by Factoring

7.4.3 Solving Trigonometric Equations with Coefficients in the Argument

7.4.4 Solving Trigonometric Equations Using the Quadratic Formula

7.4.5 Solving Word Problems Involving Trigonometric Equations

7.5.1 Identities for Sums and Differences of Angles

7.5.2 Using Sum and Difference Identities

7.5.3 Using Sum and Difference Identities to Simplify an Expression

7.6.1 Confirming a Double-Angle Identity

7.6.2 Using Double-Angle Identities

7.6.3 Solving Word Problems Involving Multiple-Angle Identities

7.7.1 Using a Cofunction Identity

7.7.2 Using a Power-Reducing Identity

7.7.3 Using Half-Angle Identities to Solve a Trigonometric Equation

8.1.1 The Law of Sines

8.1.2 Solving a Triangle Given Two Sides and One Angle

8.1.3 Solving a Triangle (SAS): Another Example

8.1.4 The Law of Sines: An Application

8.2.1 The Law of Cosines

8.2.2 The Law of Cosines (SSS)

8.2.3 The Law of Cosines (SAS): An Application

8.2.4 Heron's Formula

8.3.1 An Introduction to Vectors

8.3.2 Finding the Magnitude and Direction of a Vector

8.3.3 Vector Addition and Scalar Multiplication

8.4.1 Finding the Components of a Vector

8.4.2 Finding a Unit Vector

8.4.3 Solving Word Problems Involving Velocity or Forces

8.5.1 Graphing a Complex Number and Finding Its Absolute Value

8.5.2 Expressing a Complex Number in Trigonometric or Polar Form

8.5.3 Multiplying and Dividing Complex Numbers in Trigonometric or Polar Form

8.6.1 Using DeMoivre's Theorem to Raise a Complex Number to a Power

8.6.3 More Roots of Complex Numbers

8.6.4 Roots of Unity

8.7.1 An Introduction to Polar Coordinates

8.7.2 Converting between Polar and Rectangular Coordinates

8.7.3 Converting between Polar and Rectangular Equations

8.7.4 Graphing Simple Polar Equations

8.7.5 Graphing Special Polar Equations

9.1.1 An Introduction to Linear Systems

9.1.2 Solving a System by Substitution

9.1.3 Solving a System by Elimination

9.2.1 An Introduction to Linear Systems in Three Variables

9.2.2 Solving Linear Systems in Three Variables

9.2.3 Solving Inconsistent Systems

9.2.4 Solving Dependent Systems

9.2.5 Solving Systems with Two Equations

9.3.1 Investments

9.3.2 Solving with Partial Fractions

9.3.3 Partial Fractions: Another Example

9.4.1 Solving Nonlinear Systems Using Elimination

9.4.2 Solving Nonlinear Systems by Substitution

9.5.1 An Introduction to Matrices

9.5.2 The Arithmetic of Matrices

9.5.3 Multiplying Matrices by a Scalar

9.5.4 Multiplying Matrices