1912·14.9 MB·English

J ^' ^56 iS/ ELEMENTAEY ^/^^^ TEIGONOMETRY BY H. S. HALL, M.A., FOEMERLY SCHOLAR OF CHRIST's COLLEGE, CAMBRIDGE. AND S. R. KNIGHT, B.A., M.B., Ch.B., FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE. FOURTH EDITION, REVISED AND ENLARGED. UonUon MACMILLAN AND CO., Limited NEW YORK THE MACMILLAN COMPANY : 1906 AU niihtif renerved. First Edition, 1893. Second Edition, 1S95, 1S96, 1897. Third Edition, 1898, 1899, 1901, 1902, 1904. Fonrth Editioti, revised and enlarged, 1905 Reprinted igo6 PREFACE. Thk distinctive features of tlie Fourth Exlition are : PAGK (1) Practical Exercises in constructing angles with given ratios, and in finding the trigonometrical ratios of given angles. 7j^ (2) The Use of Four-Figure Tables of sines, cosines, and tangents 29 . . . ^^ (3) Easy Problems requiring Four-Figure Tables -18,^ (i) Graphs of the Trigonometrical Functions 79^ (5) A set of Easy Miscellaneo.us E.xam.ple.s on Chapters xi and xii 1^2^ (6) The Use of Four-Figure Logarithms and Antilogarithms . " . . . . 163v ..... (7) Solution of triangles with Four-Figure Logarithms 183^ (8) Four-Figure Tables of Ijogarithms, Anti- .... logaritlnns, Natural and Logarithmic .374 Functions The Tables uf Logarithms and Antilogarithm.s have been taken, with slight inodifications, from those published by the Board of Education, South Kensington. The Four-Figure Tables of Natural and Logarithmic Functions have been reduced from Seven-Figure Tables. For these I aui greatly indebted to Mr Frank Castle, who kindly undertook the laborious task of a special compilation for this book. (9) An easyfirst course has been mapped out enabling teachers to postpone, if they wish, all but the easier kinds of identities and transformations, so as to reach the more practical parts of the subject as early as possible. All the special features of earlier editions have Vjeen retained, and it is hoped that the present additions will satisfy all modern requirements. H. S. HALL. August 1905. SUGGESTIONS FOR A FIRST COURSE. In the iirst eighteen chapters an asterisk has been jjlaeed before all articles and sets of examples which may conveniently be omitted from a tirst course. For those who wish to postpone the haixler identities and transformations, so as to reach practical work with Four-Figure Logarithms at an eai'lier stage, the following detailed coiirse is recommended. Chaps. I—HI, Arts. 1—30, 32, 33. [Omit Art. 31, Examples in. b.] — Chaps. IV IX. [Postpone Chaps, xi and xii.] Chaps. XIII—XV, Arts. 137—170, 182^—182^. [Omit Seven-Figure Tables, Arts. 171—182.] Chaps. XI, XII. [Omit Arts. 127, 136, Examples xi.f. and XII. e.] Chap. XVI, Arts. 183—187, 197v—197d. [Omit Solu- — tions with Seven-Figure Tables, Arts. 188 197.] Chaps. XVII, XVIII, Arts. 198—218. From this point the omitted sections must be taken at the discretion of the Teacher. vn Digitized by tine Internet Arciiive in 2010 witii funding from University of Britisii Columbia Library http://www.archive.org/details/elementarytrigoOOhall CONTENTS. Chapter I. measur.emen.t u.v an.gles.. .... Page Definitionof Angle 1 Sexagesimaland Centesimal Measures . . . , . 2 Formula —=—-. . . . . . . , . . £ Chapter II, trigonometrical ratios. ... ..... Definitions of Eatio and Comniensuiable Quantities 5 Definitions ofthe Trigonometrical Eatios 6 Sine and cosine are less than unity, secant and cosecant are greaterthan unity, tangent andcotang.ent.areu.nres.tric.ted. 7 The trigonometrical ratios are independent of the lengths of the lineswhich include the angle 9 Definition offunction 10 Chapter III. relations between the ratios. Thereciprocalrelations ... 12 Tangentandcota.ngen.tin.terms.of .sine.and c.osin.e .. 13 Sine-cosine, tangent-secant, cotangent-cosecantfoiinulse . . 14 Easy Identities 16 Each ratio canbe expressedin terms of any of the others . 21 Chapter IV. trigonometrical ratios of certain angles. ...... Trigonometrical Ratios of 45°, 60°, .S0° 24 Definition of complementaryangles 27 ...... The Use ofTables ofNaturalFun.ctio.ns ..... 'J".l\ EasyTrigonometrical Equations ?{0 Miscellaneous Examples. A. 32 CONTENTS. Chapter V. solution of right-.angl.ed .TRiA.NOLE.a. . PAGE Case I. When two sides aregiven Case II. When one s.ide a.nd o.neac.ute.angle.are.give.n .. 36 Case of triangle considered as sum or difference of two right- angledtriangles 38 Chapter VI. easy problems. Angles ofElevationandDepression 41 The Mariner's Compass 45 Chapter VII. radi.an .or .ciro.ula.r me.asure. Definition of Radian 49 ...... Circumference of circle=27r (radius) . 50 Allradiansare equal 51 ITradians=2 right angles=:180 degrees 52 Radian co—ntains 57'2958 degrees 52 Formula =- 53 Values of the functions of —4, 3—,b- . 55 Ratios of thecomplementary angle —- . 56 Radian measure of angles of aregularpolygon . 56 -Rr^ad-i,.an measureof„ an angl,e=subtreandd=-i.iunsgarc 58 Radian andCircularMeasures areequivalent 58 Miscellaneous Examples. B. . . . . 61 Chapter VIII. ratios of angles op any srAGNiTUDE. ......... Convention of Signs (1) for line, (2) for plane surface, (3) for angles 64 Definitionsofthe trigonometrical.rati.os o.fan.yau.gle. 66 Signsof thetrigonometrical ratios in the fourquadrants 68 Definition of Coterminal Angles 68 ....... The fundamental formula' of Chap. iii. are true for all ahu of the augle 70 . The ambiguityof sign in cos/I— ^.sJl-sin'^A can be removed whenA isknown 71

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