Flashback For Oblique Triangles If you are given: AAS or ASA- SSS-

Slides:

Advertisements

Similar presentations

Notes 6.6 Fundamental Theorem of Algebra

Advertisements

Pre-Calculus Chapter 6 Additional Topics in Trigonometry.

Complex Numbers.

Chapter 6 – Trigonometric Functions: Right Triangle Approach

Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Complex Numbers Standard form of a complex number is: a + bi. Every complex polynomial function.

Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.

Zeros of Polynomials PolynomialType of Coefficient 5x 3 + 3x 2 + (2 + 4i) + icomplex 5x 3 + 3x 2 + √2x – πreal 5x 3 + 3x 2 + ½ x – ⅜rational 5x 3 + 3x.

9.7 Products and Quotients of Complex Numbers in Polar Form

Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.

Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.

Sec. 6.6b. One reason for writing complex numbers in trigonometric form is the convenience for multiplying and dividing: T The product i i i involves.

Complex Zeros; Fundamental Theorem of Algebra

Zeros of Polynomial Functions Section 2.5 Page 312.

6.6 The Fundamental Theorem of Algebra

1 Using the Fundamental Theorem of Algebra.  Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x 2 +

Ch 2.5: The Fundamental Theorem of Algebra

DeMoivre’s Theorem The Complex Plane. Complex Number A complex number z = x + yi can be interpreted geometrically as the point (x, y) in the complex plane.

Lesson 2.5, page 312 Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros, and to use Descartes’

6.1 Law of Sines. Introduction Objective: Solve oblique triangles To solve: you must know the length of one side and the measures of any two other parts.

Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-1 Chapter 7 Applications of Trigonometry.

Notes Over 8.1 Solving Oblique Triangles To solve an oblique triangle, you need to be given one side, and at least two other parts (sides or angles).

Section 8.1 Complex Numbers.

Section 3.3 Theorems about Zeros of Polynomial Functions.

Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.

2.5 The Fundamental Theorem of Algebra Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial. Students.

Section 4.2 – The Law of Sines. If none of the angles of a triangle is a right angle, the triangle is called oblique. An oblique triangle has either three.

2.3 Real and Non Real Roots of a Polynomial Polynomial Identities Secondary Math 3.

3.6 Complex Zereos. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra says that every polynomial with complex coefficients must have.

6.5 Theorems About Roots of Polynomial Equations

Remainder and Factor Theorems

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.

11.4 Roots of Complex Numbers

Copyright © Cengage Learning. All rights reserved. 6 Additional Topics in Trigonometry.

CHAPTER 7 SECTION 1 ROOTS AND RADICAL EXPRESSIONS Algebra 2 Notes ~ April 10, 2009.

Lesson 6.5 Trigonometric Form of Complex Numbers.

Applications of Trigonometric Functions

Use Law of Sines and the Law of Cosines to solve oblique triangles Find areas of Oblique triangles Represent vectors as directed line segments Perform.

Algebra 2 List all the integer factors for the number below: 36.

Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.

Roots & Zeros of Polynomials part 1

If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section 6.6 The Fundamental Theorem of Algebra

When solving #30 and #34 on page 156, you must “complete the square.”

Solving Polynomial Functions

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Law of Cosine Chapter 8.3.

Essential question: How do I solve oblique triangles?

Law of Sines What You will learn:

50 a 28.1o Warm-up: Find the altitude of the triangle.

Complex Numbers: Trigonometric Form

De Moivre’s Theorem and nth Roots

De Moivre’s Theorem and nth Roots

College Algebra Chapter 1 Equations and Inequalities

Chapter 2 notes from powerpoints

Law of Sines Notes Over If ABC is a triangle with sides a, b, c, then according to the law of sines, or.

Law of Sines AAS ONE SOLUTION SSA AMBIGUOUS CASE ASA ONE SOLUTION

Complex Numbers and i is the imaginary unit

Fundamental Thm. Of Algebra

Chapter 4 – Polynomial and Rational Functions

6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

Triangle Congruency Theorems (shortcuts)

Law of Sines. Law of Sines Non Right Triangles How is a triangle classified if none of the angles are 90? Oblique Labeling Oblique Triangles To solve.

Students, Take out your calendar and your homework

7.2 The Law of Sines.

The Law of Sines.

6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem

Presentation transcript:

Flashback For Oblique Triangles If you are given: AAS or ASA- SSS- SSA- SAS- What Law do you use to solve for the unknown pieces of information? Which one has the ambiguous cases?

Chapter 6.5 Trigonometric Form of a Complex Number

The Complex Number Imaginary Axis Real Axis

Finding the Absolute Value of a Complex Number Plot: and find the absolute value: Imaginary Axis Real Axis

Trigonometric Form Of a Complex Number Imaginary Axis Real Axis

Write in Trigonometric Form Imaginary Axis Real Axis

Try #15 page 440

Writing a Complex Number in Standard Form Write in Standard Form

Multiplying Complex Numbers in Trigonometric Form Find the product of the complex numbers

Dividing Complex Numbers in Trigonometric Form Find the Quotient of the complex numbers

Powers of Complex Numbers DeMoives Theorem If is a complex number and n is a positive integer, then

Use DeMoive’s Theorem to find

Remember that an nth degree polynomial has at least one complex zero (root) and at most n complex zeros (roots). A complex number also has n nth roots. For example, has 3 cube roots. To find them you need to use this formula: for a complex number The n distinct nth roots are given by where k = 0, 1, 2, …, n - 1

Example: Find the 4 fourth roots of . If the n distinct nth roots are given by where k = 0, 1, 2, …, n - 1