1) Trig form of a Complex # 2) Multiplying, Dividing, and powers (DeMoivre’s Theorem) of Complex #s 3) Roots of Complex #s Section 6-5 Day 1, 2 &3

1) Trig form of a Complex # 2) Multiplying, Dividing, and powers (DeMoivre’s Theorem) of Complex #s Section 6-5 Day 1& 2

Warm-Up Find the work done by a man pushing a car with 60 lbs of force at an angle 30 degrees below horizontal for 1000 feet. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

Section 6-5 Day 1 The Trigonometric form of a Complex Number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 The standard form of a complex number is z = a+bi In the complex plane, every complex number corresponds to a point. Definition: Complex Plane Example: Plot the points 3 + 4i and –2 – 2i in the complex plane. Imaginary axis Real axis 2 4 – 2 2 (3, 4) or 3 + 4i (– 2, – 2) or – 2 – 2i

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 The absolute value of the complex number z = a + bi is the distance between the origin (0, 0) and the point (a, b). Definition: Absolute Value Example: Plot z = 3 + 6i and find its absolute value. Imaginary axis Real axis 4 4 – 2 – z = 3 + 6i

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 To write a complex number a + bi in trigonometric form, let  be the angle from the positive real axis (measured counter clockwise) to the line segment connecting the origin to the point (a, b). Trigonometric Form of a Complex Number a = r cos  b = r sin  Imaginary axis Real axis b r a (a, b) 

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition: Trigonometric Form of a Complex Number The trigonometric form of a complex number z = a + bi is given by z = r(cos  + i sin  ) where a = r cos , b = r sin , The number r is the modulus of z, and  is the argument of z. Example: modulus argument

How is graphing in trig form different? In a rectangular system, you go left or right and up or down. In a trigonometric or polar system, you have a direction to travel and a distance to travel in that direction.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Write the complex number z = –7 + 4i in trigonometric form. Example: Trigonometric Form of a Complex Number Imaginary axis Real axis z = –7 + 4i °

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 You try: Write the complex number in trigonometric form. Example: Trigonometric Form of a Complex Number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 You try: Write the complex number in trigonometric form. Example: Trigonometric Form of a Complex Number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Standard Form of a Complex Number in Radians Standard form Write the complex number in standard (rectangular) form a + bi. Example:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Standard Form of a Complex Number in Degrees Write the complex number in standard form a + bi. You Try: Standard form

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: Standard Form of a Complex Number in Degrees Write the complex number in standard form a + bi. You Try:

Homework Day 1: Pg. 440, 1-47 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Section 6-5 Day 2 Multiplying, Dividing, and Powers (DeMoivre’s Theorem) of Complex #s

Homework Quiz Find the trigonometric form of the complex number: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Multiplication of Complex Numbers Multiply the 2 complex numbers. Example: Trigonometric Form of a Complex Number Hint: Write the numbers in standard form and multiply algebraically. There is an easier way

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Multiplication of Complex Numbers To multiply 2 complex numbers, you multiply the moduli and add the arguments. Example: Find the product of the complex numbers and write it in standard form: Example: Trigonometric Form of a Complex Number

You Try: Find the product of the two complex numbers. Find both the trigonometric form and standard form of the product. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Division of Complex Numbers To divide 2 complex numbers, you divide the moduli and subtract the arguments. Example: Find the quotient of the complex numbers and write it in standard form: Example: Trigonometric Form of a Complex Number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Division of Complex Numbers To divide 2 complex numbers, you divide the moduli and subtract the arguments. You Try: Find the quotient of the complex numbers and write it in standard form: Example: Trigonometric Form of a Complex Number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Example: Write (1 + i) 2 in standard form a + bi. Write (1 + i) 5 in standard form a + bi. Powers of Complex Numbers

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 To raise a complex number to a power, you can use it’s trigonometric form: Powers of Complex Numbers

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 To raise a complex number to a power, you can use it’s trigonometric form: Powers of Complex Numbers

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Raising complex numbers by powers creates a pattern: Powers of Complex Numbers

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 DeMoivre’s Theorem Definition: DeMoivre’s Theorem If z = r(cos  + i sin  ) is a complex number and n is a positive integer, then z n = [r(cos  + i sin  )] n = r n (cos n  + i sin n  )]. Example:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Example: DeMoivre’s Theorem Example: Use DeMoivre’s Theorem to write (1 + i) 5 in standard form a + bi. Convert the complex number into trigonometric form. Example continues.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 Example Continued Example continued:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 Example: DeMoivre’s Theorem You Try: Write (3 + 4i) 3 in standard form a + bi.

Homework Day 2: Pg. 441, odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32

Warm-Up Find the indicated power of the complex number by rewriting the number in trig form and using DeMoivre’s Theorem. Write the result in standard form. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33

Section 6-5 Day 3 Roots of Complex Numbers

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35 How many solutions? Difference of squares? Sum and difference of cubes?

Roots of Complex Numbers There will be as many answers as the index of the root you are looking for. –Square root = 2 answers –Cube root = 3 answers, etc. Answers will be spaced symmetrically around the circle –You divide a full circle by the number of answers to find out how far apart they are

General Process 1.The complex number must be in trig. form. 2.Take the n th root of r. All answers have the same value for r. 3.Divide theta by n to find the first angle. 4.Divide a full circle by n to find out how much you add to theta to get to each subsequent answer. 5.Write your numbers, increasing by theta/n each time.

The formula k starts at 0 and goes up to n-1 This is easier than it looks.

First, write as a root and write the radicand in trig. form i is over 1 and up 0. Therefore, 1 is the hypotenuse and theta is 0 o. Or use arctan b/a. First angle? Divide theta by n to find the first angle. How far apart will the evenly spaced angles be? The first angle is 0.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 41 The sixth roots of 1.

Example: 1. Find the 4 th root of Divide theta by 4 to get the first angle. 3.Divide 360 by 4 to determine the spacing between angles

Copyright © by Houghton Mifflin Company, Inc. All rights reserved List the 4 answers. The only thing that changes is the angle. The number of answers equals the number of roots.

You Try: Find the square roots of Write them in standard form. Remember to convert to trig form first.

You Try: Find the cube roots of z = -2+2i Again, first convert to trig form.

Homework Day 3: Pg. 442, odd, don’t need to graph them. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 46