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Homework, Page 548 (a) Complete the table for the equation and (b) plot the points. 1. r 3cos 2 θ r 0 π/4 π/2 3 0 –3 π 3 3π/4 0 5π/4 0 3π/2 –3 7π/4 0 y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1 Homework, Page 548 Draw the graph of the rose curve. State the smallest θ-interval (0 ≤ θ ≤ k) that will produce a complete graph. 5. r 3cos 2 y x The smallest -interval that will produce the entire graph is 0 2 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 2 Homework, Page 548 Match the equation with its graph without using your calculator. 9. Does the graph of r 2 2sin or r 2 2cos appear in the figure? Explain. r 0 2 2sin 0 2 r 0 2 2cos 0 0 2 2 2sin 2 4 r 2 2cos 2 2 2 r Points 0,0 and 2, 2 are on the graph, so the graph is of r 2 2cos Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 3 Homework, Page 548 Use the polar symmetry tests to determine if the graph is symmetric about the x-axis, the y-axis, or the origin.. 13. r 3 3sin r 3 3sin r 3 3sin 3 3sin r r 3 3sin 3 3sin r r 3 3sin 3 3sin r r 3 3sin 3 3sin r Symmetric about the y -axis Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Homework, Page 548 Use the polar symmetry tests to determine if the graph is symmetric about the x-axis, the y-axis, or the origin.. 17. r 5cos 2 r 5cos 2 5 cos 2 sin 2 r 5 cos sin 5 cos r 5 cos sin 5 cos r 5 cos 2 sin 2 5 cos 2 sin 2 r 2 2 2 2 sin 2 r 2 sin 2 r 2 Symmetric about the y-axis, the x-axis, and the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Homework, Page 548 Identify the points on 0 ≤ θ ≤ 2π where maximum r-values occur. 21. r 2 3cos The maximum r -values occur for r 2 3cos at 0, 2 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Homework, Page 548 Analyze the graph of the polar curve. r 3 25. Domain: r 3 Range: Continuity: Continuous Symmetry: Symmetric about the y-axis, the x-axis, and the origin. Boundedness: Bounded Maximum r-value: r 3 None Asymptotes: Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Homework, Page 548 Analyze the graph of the polar curve. 29. r 2sin 3 Domain: Range: 2 r 2 Continuity: Continuous Symmetry: Symmetric about the y-axis. Boundedness: Bounded Maximum r-value: r 2 Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 Homework, Page 548 Analyze the graph of the polar curve. 33. r 4 4cos Domain: Range: 0 r 8 Continuity: Continuous Symmetry: Symmetric about the x-axis Boundedness: Bounded Maximum r-value: r 8 Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Homework, Page 548 Analyze the graph of the polar curve. 37. r 2 5cos Domain: Range: 3 r 7 Continuity: Continuous Symmetry: Symmetric about the x-axis Boundedness: Bounded Maximum r-value: r 7 Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10 Homework, Page 548 Analyze the graph of the polar curve. r 2 41. Domain: Range: 0 r Continuity: Continuous Symmetry: Not symmetrical Boundedness: Unbounded Maximum r-value: r Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11 Homework, Page 548 Find the length of each petal of the polar curve. 45. r 2 4sin 2 The large petals are 6 units long and the short petals are 2 units long. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12 Homework, Page 548 Select the two equations whose graphs are the same curve. Then, describe how the paths are different as θ increases from 0 to 2π. 49. r1 1 3sin , r2 1 3sin , r3 1 3sin The graphs of r1 and r2 are the same curve. The graph of r1 starts with the outside portion and ends with the inside portion. The graph of r2 starts with the inside portion and ends with the outside portion. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 13 Homework, Page 548 (a) Describe the graph of the polar equation, (b) state any symmetry the graph possesses, and (c) state the maximum rvalue, if it exists. 2 r 2sin 2 sin 2 53. (a) The graph of the polar equation has two large petals and two small petals. (b) The graph is symmetric about the origin. (c) The maximum r-value is 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14 Homework, Page 548 57. Analyze the graphs of the polar equations r a cos n and r a sin n when n is an even integer. Domain: All real numbers. Range: a r a Continuity: Continuous Symmetry:Symmetric about the x-axis, y-axis, and origin. Boundedness: Bounded Maximum r-value: a Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 15 Homework, Page 548 61. A polar curve is always bounded. Justify your answer. False The spiral curve, the graph of the polar equation r = θ is unbounded. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 16 Homework, Page 548 65. Which of the following is the maximum r-value for r = 2 – 3 cos θ? a. 6 r 2 3cos 1 cos 1 b. 5 3 1 3 2 3 5 3 1 3 2 3 1 5 1 5 c. 3 d. 2 e. 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17 Homework, Page 548 69. The graphs of r1 3sin 5 2 and r2 3sin 7 2 may be called rose curves. a Determine the smallest -interval that will produce a complete graph of r1; of r2 . It took an inteval of 0 4 to complete each graph. b How many petals does each graph have? Graph r1 has 10 petals and graph r2 has 14 petals. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18 6.6 De Moivre’s Theorem and nth Roots Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review 1. Write the roots of the equation x 12 6 x in a bi form. 2 2. Write the complex number 1 i in standard form a bi. 3 3. Find all real solutions to x 27 0. Find an angle in 0 2 which satisfies both equations. 3 1 3 and cos 2 2 2 2 5. sin and cos 2 2 4. sin Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 20 Quick Review Solutions 1. Write the roots of the equation x 12 6 x in a bi form. 2 3 3i, 3 3i 2. Write the complex number 1 i in standard form a bi. 3 2 2i 3. Find all real solutions to x 27 0. x 3 Find an angle in 0 2 which satisfies both equations. 3 1 3 and cos 5 / 6 2 2 2 2 5. sin and cos 5 / 4 2 2 4. sin Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 21 What you’ll learn about The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of Complex Numbers Roots of Complex Numbers … and why The material extends your equation-solving technique to include equations of the form zn = c, n is an integer and c is a complex number. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22 Complex Plane Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23 Addition of Complex Numbers Two complex numbers, z1 a1 b1i and z2 a2 b2i may be added by separately adding their real and imaginary parts z1 z2 a1 a2 b1 b2 i. This addition in the complex plane is the same operation as vector addition in the real Cartesian plane. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24 Absolute Value (Modulus) of a Complex Number The absolute value or modulus of a complex number z a bi is | z || a bi | a b . In the complex plane, | a bi | is the distance of a bi from the origin. 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Slide 6- 25 Graph of z = a + bi Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 26 Trigonometric Form of a Complex Number The trigonometric form of the complex number z a bi is z r cos i sin where a r cos , b r sin , r a b , 2 2 and tan b / a. The number r is the absolute value or modulus of z, and is an argument of z. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27 Example Finding Trigonometric Form Find the trigonometric form with 0 2 for the complex number 1 3i. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 28 Product and Quotient of Complex Numbers Let z1 r1 cos1 i sin 1 and z2 r2 cos 2 i sin 2 . Then 1. z1 z2 r1r2 cos 1 2 i sin 1 2 . 2. z1 r1 cos 1 2 i sin 1 2 , r2 0. z2 r2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 29 Example Multiplying Complex Numbers Express the product of z1 and z2 in standard form. z1 4 cos i sin , z2 2 cos i sin 4 4 6 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 30 Example Dividing Complex Numbers Express the quotient of z1 z2 in standard form. z1 6 cos i sin , z2 3 cos i sin 3 3 5 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 31 A Geometric Interpretation of z2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 32 De Moivre’s Theorem Let z r cos i sin and let n be a positive integer. Then z r cos i sin r n cos n i sin n . n n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 33 Example Using De Moivre’s Theorem Find 1+ 3i 4 using De Moivre's theorem. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 34 nth Root of a Complex Number A complex number v a bi is an nth root of z if v n z. If z 1, then v is called an nth root of unity. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 35 Finding nth Roots of a Complex Number If z r cos i sin , then the n distinct complex numbers 2 k 2 k r cos i sin , where k 0,1, 2,.., n 1, are n n the nth roots of the complex number z. n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 36 Example Finding Cube Roots Find the cube roots of 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 37 Homework Homework Assignment #8 Review Section: 6.6 Page 559, Exercises: 1 – 69 (EOO), 81, 83 Quiz next time Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 38