Projects - Cengage Learning

More documents

Recommendations

Info

C-70 Appendix C EXAMPLE 3 Values of Some Expressions Involving the Second Version of the Inverse Secant p You can show that Sec 1 2 3, Sec[Sec 1 (5)] 5, and Sec 1 (Sec 2 as in Example 1. However, Arcsec(12) the unique number in [0, p 3 ) 2 5p whose secant is 12 3p 2 ) 3 (p 2 , p] 4 , not as in Example 1. 4 Note: Sec 1 x sec 1 x for x 1, but Sec 1 x 2p sec 1 x for x 1. EXAMPLE 4 Evaluating Expressions Involving the Second Version of the Inverse Secant sin[Sec 1 ( 5 2)] 1215 as in Example 2(a). Or, by the previous note, œ„„ 55 8 ~ ¨ 3 y ¨ x 8 sin c Sec 1 a 5 2 bd sin c sec1 a 5 121 bd 2 5 But, Sec 1 ( 3) is in ( p 2, p], sec 1 ( 3) is in [p, 3p and neither equals p. (Why?) So and in fact Z tan [sec 1 8 tan[Sec 1 8 Sec 1 8 ( 3) sec 1 2 ), 8 ( 3), ( 3)] ( 3)]. To evaluate let u Sec 1 8 tan[Sec 1 8 ( 3)], ( 3) and let ũ be its reference angle. Then u is in ( p 2, p] and Sec u 8 3. Using the right triangle labeled in Figure E, we have opposite ũ Why is adjacent 155 tan[Sec 1 8 ( 3)] tan u tan 3 tan tan ˜? 8 Figure E Alternatively, using identities, sec u 8 3, so 155 tan u 2sec 2 u 1 2( 83) 2 1 3 Why the negative square root? Exercises In each of these exercises the secant and the Secant are the restricted versions defined in this project, so the inverse secant and the inverse Secant are the corresponding inverse functions. 1. Evaluate the following quantities: (a) sec 1 (0) (c) arcsec(1) (e) arcsec 12 (b) sec 1 1 2132 (d) sec[sec 1 (10)] (f) sec 1 (sec 4) 2. Evaluate the following quantities: (a) (b) sin[arcsec(5)] (c) tan3sec 1 7 cos3sec 1 1 6 524 1 3 24 3. Evaluate the following quantities: (a) Sec 1 (1) (c) Arcsec(2) (e) Sec[Sec 1 (15)] (b) Sec 1 (Sec 2) (d) (f) tan3Sec 1 15 sin3Sec 1 1 13 5 24 1 7 24 4. Prove Sec 1 x sec 1 x for x 1 and Sec 1 x 2p sec 1 x for x 1 5. Use the result in Problem 4 to show that tan3sec 1 8 tan3Sec 1 8 1 324 1 324 1553 6. Explain why each of the following statements is true: (a) sec[Sec 1 (3)] 3 (e) Sec 1 3sec1 4 324 4 (b) sec[Sec 1 (3)] is undefined (f) sec 1 3Sec 3p (c) Sec[sec 1 (8)] 8 (g) sec 1 4 4 5p 3 4 (Sec 2) 2p 2 (d) Sec[sec 1 (8)] is undefined
Appendix C C-71 PROJECT Snell’s Law and an Ancient Experiment The “lifting” effect produced by refraction is the basis for one of the earliest recorded experiments in optics, one which was known to the ancient Greeks. A coin is placed in the bottom of an empty vessel and the eye of an observer is placed in such a position that the coin is hidden below the edge of the vessel. If water is poured into the vessel the coin appears to rise and come into view. —Francis Weston Sears in Optics (Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1958) In this group project you will learn some of the basic ideas of geometrical optics, a subject first studied by the ancient Greeks, advanced by artists and mathematicians of the Renaissance, and further developed by mathematicians and physicists from the seventeenth through the twentieth and into the twentyfirst century. Major contributors include Archimedes, Descartes, Fermat, Huygens, Newton, Gauss, Hamilton, Fresnel, Einstein, and Born. We now discuss some of the basic principles. The first one is the notion of a light ray. When traveling through a uniform transparent medium (such as a vacuum, water, plastic, or glass) light travels along straight line paths called light rays. The second principle is that light rays bend when passing from one medium to another, for example, when passing from water to air. The amount of bending is determined by the Law of Refraction, usually referred to as Snell’s Law after its discoverer, Willebrod Snell (1580–1626). When a light ray traveling in a transparent medium intersects the boundary with another transparent medium, its direction changes as indicated in Figure A. Incident ray Normal line Angle of incidence Boundary Angle of refraction ˙ First medium: index of refraction n Point of incidence Second medium: index of refraction nª ˙ª Figure A Refracted ray Using the terminology introduced in Figure A we can state Snell’s Law in two parts: I. The incident ray, the refracted ray and the line normal (or perpendicular) to the boundary at the point of incidence all lie in the same plane. II. The angle of incidence and the angle of refraction are related by the equation n sin f n¿ sin f¿ The index of refraction of a vacuum is defined to be 1. For other materials it may be determined empirically by precise measurement of angles. Air has
Page 1 and 2:
APPENDIX C Title and Page Number Pr
Page 3 and 4:
Appendix C C-3 MINI PROJECT Discuss
Page 5 and 6:
Appendix C C-5 MINI PROJECT Thinkin
Page 7 and 8:
Appendix C C-7 1. equilateral trian
Page 9 and 10:
Appendix C C-9 MINI PROJECT Flying
Page 11 and 12:
Appendix C C-11 MINI PROJECT An Ine
Page 13 and 14:
Appendix C C-13 (a) The two data po
Page 15 and 16:
Appendix C C-15 and http://www.svob
Page 17 and 18:
Appendix C C-17 For Figure D we can
Page 19 and 20: Appendix C C-19 MINI PROJECT A Grap
Page 21 and 22: Appendix C C-21 MINI PROJECT Who Ar
Page 23 and 24: Appendix C C-23 MINI PROJECT 1 How
Page 25 and 26: Appendix C C-25 either the quadrati
Page 27 and 28: Appendix C C-27 PROJECT The Least-S
Page 29 and 30: Appendix C C-29 Then, for any line
Page 31 and 32: Appendix C C-31 PROJECT Finding Som
Page 33 and 34: Appendix C C-33 3. When a person co
Page 35 and 36: Appendix C C-35 2. Use part (b) of
Page 37 and 38: Appendix C C-37 PROJECT Coffee Temp
Page 39 and 40: Appendix C C-39 MINI PROJECT More C
Page 41 and 42: Appendix C C-41 The calculations co
Page 43 and 44: Appendix C C-43 PROJECT Transits of
Page 45 and 46: Appendix C C-45 E A ¨ V ¨ Bª d S
Page 47 and 48: Appendix C C-47 PROJECT A Linear Ap
Page 49 and 50: Appendix C C-49 Continuing with the
Page 51 and 52: Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
Page 61 and 62: Appendix C C-61 PROJECT The Design
Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
Page 67 and 68: Appendix C C-67 PROJECT Inverse Sec
Page 69: Appendix C C-69 Alternatively, usin
Page 73 and 74: Appendix C C-73 vertically downward
Page 75 and 76: 0 0 0 Appendix C C-75 PROJECT Vecto
Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
Page 81 and 82: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
Page 83 and 84: Appendix C C-83 PROJECT Parameteriz
Page 85 and 86: Appendix C C-85 It is worth noting
Page 87 and 88: Appendix C C-87 6. Show that the li
Page 89 and 90: Appendix C C-89 2. An altitude of a
Page 91 and 92: Appendix C C-91 PROJECT The Leontie
Page 93 and 94: Appendix C C-93 n x units of that
Page 95 and 96: Appendix C C-95 him/herself through
Page 97 and 98: Appendix C C-97 PROJECT The Leontie
Page 99 and 100: Appendix C C-99 As in the solution
Page 101 and 102: Appendix C C-101 MINI PROJECT 2 Con
Page 103 and 104: Appendix C C-103 PROJECT Using Hype
Page 105 and 106: Appendix C C-105 PROJECT Two Method
Page 107 and 108: Appendix C C-107 (b) Use Method 1 t
Page 109 and 110: Appendix C C-109 MINI PROJECT An Un
Page 111 and 112: Appendix C C-111 PROJECT More on Su
Page 113 and 114: Appendix C C-113 SOLUTION 25 25 a (
Page 115: Appendix C C-115 4. Simplify the fo
show all

Projects - Cengage Learning

Create successful ePaper yourself

Delete template?

Save as template?