Algebra-Trigonometry-CynthiaYoung-3ed95.pdf - c08d.qxd 10:49 AM Page 911 8.6 Polar(Trigonometric Form of Complex Numbers 911 To convert from polar to

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8.6 Polar (Trigonometric) Form of Complex Numbers 911 To convert from polar to rectangular form, simply evaluate the trigonometric functions. EXAMPLE 4 Converting from Polar to Rectangular Form Express in rectangular form. Solution: Evaluate the trigonometric functions exactly. Distribute the 4. Simplify. YOUR TURN Express in rectangular form. z = 2(cos 210° + i sin 210°) z = - 2 + 2 1 3 i z = 4 a - 1 2 b + 4 a 1 3 2 b i b cos 120° + i sin 120° a z = 4 z = 4(cos 120° + i sin 120°) 1 2 1 3 2 EXAMPLE 5 Using a Calculator to Convert from Polar to Rectangular Form Express in rectangular form. Round to four decimal places. Solution: Use a calculator to evaluate the trigonometric functions. Simplify. YOUR TURN Express in rectangular form. Round to four decimal places. z = 7(cos 217° + i sin 217°) z L - 0.9767 + 2.8366 i b cos(109°) + i sin(109°) a z = 3 z = 3(cos 109° + i sin 109°) 0.325568 0.945519 SECTION 8.6 and and 0 2 . It is important to note in x Z 0 tan u = y x , In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis. We can express complex numbers in either rectangular form, z x iy , or polar form, z r (cos i sin ). The modulus of a complex number, z x iy , is given by . To convert from rec- ƒ z ƒ = 2 x 2 + y 2 SUMMARY tangular to polar form, we use the relationships r = 2 x 2 + y 2 which quadrant the point lies. To convert from polar to rectangular form, simply evaluate the trigonometric functions. x = r cos u and y = r sin u ¯˚˘˚˙ Technology Tip Express in rectangular form. z = 4(cos 120° + i sin 120°) Answer: z = - 1 3 - i Answer: z L - 5.5904 - 4.2127 i Technology Tip Express in rectangular form. z = 4(cos 109° + i sin 109°) ENTER ENTER 6: Rect CPX MATH ) ) 109 sin . 2nd ) 109 cos ( 3 ¯˘˙ ¯˘˙ ¯˚˘˚˙
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912 CHAPTER 8 Additional Topics in Trigonometry In Exercises 1–8, graph each complex number in the complex plane. 1. 2. 3. 4. 5. 6. 7 7. 8. In Exercises 9–22, express each complex number in polar form. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. In Exercises 23–38, use a calculator to express each complex number in polar form. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. In Exercises 39–48, express each complex number in rectangular form. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. In Exercises 49–58, use a calculator to express each complex number in rectangular form. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. A PPLICATIONS - 4 c cos a 15 p 11 b + i sin a 15 p 11 bd - 2 c cos a 3 p 5 b + i sin a 3 p 5 bd 2 c cos a 4 p 7 b + i sin a 4 p 7 bd 3 c cos a 11 p 12 b + i sin a 11 p 12 bd - 5(cos 320° + i sin 320°) - 7(cos 140° + i sin 140°) 6(cos 250° + i sin 250°) 3(cos 100° + i sin 100°) 4(cos 35° + i sin 35°) 5(cos 295° + i sin 295°) 2 c cos a 5 p 6 b + i sin a 5 p 6 bd 1 2 c cos a p 4 b + i sin a p 4 bd 1 3 (cos 330° + i sin 330°) 1 3 (cos 150° + i sin 150°) - 4(cos 210° + i sin 210°) - 4(cos 60° + i sin 60°) 3(cos 270° + i sin 270°) 2(cos 315° + i sin 315°) 2(cos 135° + i sin 135°) 5(cos 180° + i sin 180°) 1.78 - 0.12 i 4.02 - 2.11 i - 4 1 5 3 + 1 5 2 i - 2 1 3 - 1 5 i 1.8 - 0.9 i 5.1 + 2.3 i - 5 8 - 11 4 i - 1 2 + 3 4 i - 3 + 4 i 8 - 6 i 24 + 7 i - 5 + 12 i - 4 - 3 i - 6 + 5 i 2 + 3 i 3 - 7 i - 6.48 + 0 i 5.32 - 0 i 1 6 - 1 6 i - 1 2 - 1 2 i - 2 + 0 i 3 + 0 i - 1 3 + i 1 3 - 3 i 1 5 - 1 5 i - 4 + 4 i - 3 - 1 3 i 1 + 1 3 i 2 + 2 i 1 - i - 5 i - 3 i 2 - 3 - 2 i - 2 - 4 i 3 + 5 i 7 + 8 i SKILLS EXERCISES SECTION 8.6
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8.6 Polar (Trigonometric) Form of Complex Numbers 913 59. Resultant Force.
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