31151029036484.pdf

More documents

Recommendations

Info

$ï V \ - JScholarship - Johns Hopkins University$

240 SPHERICAL TRIGONOMETRY, 150. CASE II. C = 90° and c constant. d A sin 2 A da d~B^~ sin 2B ~dT d a tan a d b (TA. ~ tan A d B da sin 2 a d b d B 2 cot 5 (^ ^ cot a cot b tan b tan B sin 2 b 2 cot J. 151. CASE III. C = 90° and A constant. d c tun c djz_ _ sin 2 a Ta ^ tana a
APPROXIMATE SOLUTION OP SPHERICAL TRIANGLES. 241 CHAPTER VII. APPROXIMATE SOLUTION OF SPHERICAL TRIANGLES IN CERTAIN CASES. 154. WHEN some of the parts of the triangle are small, or nearly 90°, or nearly 180°, approximate solutions may be employed with advantage. These are generally found by means of series. 155. In a spherical right triangle (the right angle being C), given A and c, to find b We have tan J = cos A tan c (288) which is of the form in PL Trig. (493), and may therefore be developed by (495) and (496) by putting x = b, y = c, p = cos A, whence p — 1 1 — cos A , , g = I T = — rn i = — tan' J A p + \ 1 -4- cos ^1 2 and (495) and (496) become, [taking n = 0 in (495), and n = 1 in (496)], i = c — tan' \Aa\n2c+\ tan' J ^ sin 4 c — &c. (289) b = TT—c 4- cot' J 4 sin 2 0 — J oof .} A sin 4 o 4- &c. (290) If .4 is small, cos A is nearly equal to unity, and b exceeds c by a small quantity which is approximately found by one or more terms of the series (289). If A is nearly 180°, or cos^ nearly = — 1, S exceeds^-—c by a small quantity, which is found by (290). For examples of the mode of computation, see PI. Trig. Art. 255. 156. Althougli these solutions are termed approximate, it must not be inferred that they are less accurate in practice tlian the direct solution of (288) by the tables ; for the logarithmic tables are themselves only approximate, and the neglect of the higher powers in such series as (289) and (290) may involve a less theoretical error than the similar neglect of the higher powers in the series by which the tables are computed. In the examples of PI. Trig. Art. 255, the thousandths of a second were found with accuracy, which could not have been effected by a direct solution with less than eight decimal places in the logarithms. These considerations lead to the frequent employment of approximate solutions in astronomy. 157. If A and b are given, to find c, we have tan c = sec J. tan 5 whicli is reduced to PI. Trig. (493), by putting z •== c, y = b, p = sec A, and the series -will be sec.4 — 1 1 — 003^ , „, , secA + \ l-|-cos4. ^ c = b + tan' J 4 sin 2 i 4- J tan' J A sin 4 6 -f &c. (291) c =!r—i — cot' J 4 sin 2 5 — J cot* I A ainib — &o. (292). 31 V
Page 2:
a A « LIBRARY JC^HNS HOPKINS UNIVE
Page 8 and 9:
Q./\ 6-^1 C f Entered according to
Page 10 and 11:
4 PKEFACE. Cagnoli's Table has been
Page 12 and 13:
6 CONTENTS. CHAPTER VIXX. Pict SOLU
Page 15 and 16:
PAET I. PLANE TEIGONOMETET. CHAPTER
Page 17 and 18:
MEASURES OF AECS. 11 other instrume
Page 19 and 20:
MEASURES OF ARCS. 13 180° ^ = 3-44
Page 21 and 22:
SINES, TANGENTS, AND SECANTS. 1,5 T
Page 23 and 24:
FUNDAMENTAL FORMULAE. [7 functions
Page 25 and 26:
FUNDAMENTAL POEMULjE. 19 The triang
Page 27 and 28:
FUNDAMENTAL FOEMUL^. 21 Then the tr
Page 29 and 30:
FUNCTIONS OF ANGULAR MAGNITUDE. 23
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
GENERAL FORMULA 81 CHAPTER rV. GENE
Page 39 and 40:
GENERAL FORMUL.*;. 61. Divide (36),
Page 41 and 42:
GENERAL FOEMULIE. 35 As these expre
Page 43 and 44:
FOEMULiE FOR MULTIPLE ANGLES. 37 If
Page 45 and 46:
RELATIONS OF THREE ANGLES. 39 The q
Page 47 and 48:
INVERSE TRIGONOMETRIC FUNCTIONS. 41
Page 49 and 50:
TRIGONOMETRIC TABLES. 43 CHAPTER V.
Page 51 and 52:
TABLE OF TRIGONOMETRIC SINES. 45 re
Page 53 and 54:
CONSTEUCTION OF TEIGONOMETEIC TABLE
Page 55 and 56:
CONSTRUCTION OF TRIGONOMETRIC TABLE
Page 57 and 58:
SOLUTION OF PLANE RIGHT TEIANGLES.
Page 59 and 60:
SOLUTION OF PLANE RIGHT TRLANGLES.
Page 61 and 62:
ADDITIONAL FORMULA FOE EIGHT TRIANG
Page 63 and 64:
FOEMULiE FOE OBLIQUE TRIANGLES. 57
Page 65 and 66:
FORMULAE FOR OBLIQUE TRIANGLES. 59
Page 67 and 68:
FORMULA FOR OBLIQUE TRIANGLES. 61 1
Page 69 and 70:
FOEMULiE FOE OBLIQUE TELINGLES. 63
Page 71 and 72:
SOLUTION OP OBLIQUE TRIANGLES. 65 2
Page 73 and 74:
and the two values of e will be SOL
Page 75 and 76:
SOLUTION OF OBLIQUE TEIANGLES. 69 2
Page 77 and 78:
SOLUTION OF OBLIQUE TRIANGLES. 7] t
Page 79 and 80:
SOLUTION OF OBLIQUE TRIANGLES. 73 E
Page 81 and 82:
PROBLEMS RELATING TO PLANE TRIANGLE
Page 83 and 84:
Page 85 and 86:
PROBLEMS EELATING TO PLANE TEIANGLE
Page 87 and 88:
PEOBLEMS EELATING TO PLANE TEIANGLE
Page 89 and 90:
Page 91 and 92:
SOLUTION OF TRIGONOMETRIC EQUATIONS
Page 93 and 94:
SOLUTION OP TRIGONOMETRIC EQUATIONS
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
EQUATIONS OF THE SECOND AND THIRD D
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
DIFFERENCES AND DIFFERENTIALS. 101
Page 109 and 110:
DIFFERENCES AND DIFFERENTIALS. 103
Page 111 and 112:
DIFFERENCES AND DIFFERENTIALS OF PL
Page 113 and 114:
DIFFERENCES OF PLANE TRUNQLES. 107
Page 115 and 116:
DIFFERENCES OP PLANE TRIANGLES 109
Page 117 and 118:
DIFFERENTIAL VARIATIONS OF PLANE TR
Page 119 and 120:
DIFFERENTIAL VARIATIONS OF PLANE TR
Page 121 and 122:
TRIGONOMETRIC SERIES. ^15 CHAPTER X
Page 123 and 124:
TRIGONOMETRIC SERIES. 117 209. To d
Page 125 and 126:
Substituting the values from (408)
Page 127 and 128:
TRIGONOMETRIC SERIES. 12I simpler t
Page 129 and 130:
Developing these logs, by the known
Page 131 and 132:
TRIGONOlVfETRIC SERIES. 125 For exa
Page 133 and 134:
EXPONENTIAL FORMUL.^. 127 CHAPTER X
Page 135 and 136:
EXPONENTIAL FORMULA. 129 221. Moivr
Page 137 and 138:
TRINOMIAL OR QUADRATIC FACTORS. 131
Page 139 and 140:
TRINOMIAL OR QUADRATIC FACTORS. 13i
Page 141 and 142:
MULTIPLE ANGLES. 135 CHAPTER XV. TR
Page 143 and 144:
MULTIPLE ANGLES. 137 (which must no
Page 145 and 146:
MULTIPLE ANGLES. 139 In these formu
Page 147 and 148:
MULTIPLE ANGLES. 1-11 in which m be
Page 149 and 150:
MULTIPLE ANGLES 143 If the denomina
Page 151 and 152:
MULTIPLE ANGLES. 145 253. The serie
Page 153 and 154:
which is reduced to (493) by puttin
Page 155 and 156:
PAET II. SPHERICAL TEIGONOMETllY. C
Page 157 and 158:
GENERAL FORMULA. l&l Again, B'O' si
Page 159 and 160:
GENERAL FORMULA. 153 2d. In the tri
Page 161 and 162:
we find the first of the following
Page 163 and 164:
then GENERAL FORMULAE. Let s denote
Page 165 and 166:
GENERAL FORMULA. 159 .._oo^{S-B)coa
Page 167 and 168:
25. Gauss's Theorem. If GENERAL FOR
Page 169 and 170:
30 We have also from (35) and (37)
Page 171 and 172:
ADDITIONAL rORMUL.ai. 165 sin /S =
Page 173 and 174:
SOLUTION OF SPHERICAL RIGHT TRIANGL
Page 175 and 176:
Page 177 and 178:
SOLUTION OP SPHERICAL RIGHT TRIANGL
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
QUADRANTAL AND ISOSCELES TRIANGLES.
Page 185 and 186:
SOLUTION OF SPHERICAL OBLIQUE TRIAN
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196: SOLUTION OF SPHERICAL OBLIQUE TRIAN
Page 209 and 210: SOLUTION OF SPHE.IICAL OBLIQUE TRIA
Page 219 and 220: SOLUTION OP SPHERICAL OBLIQUE TRIAN
Page 221 and 222: SOLUTION OF THE GENERAL SPHERICAL T
Page 223 and 224: SOLUTION OP THE GENERAL SPHERICAL T
Page 235 and 236: AREA OP A SPHERICAL TRIANGLE. 229 C
Page 237 and 238: AREA OF A SPHERICAL TRLANGLE. 281 1
Page 239 and 240: By (43), DIFFERENCES OP SPHERICAL T
Page 241 and 242: DIFFERENCES OF SPHERICAL TRIANGLES.
Page 243 and 244: DIFFERENCES OF SPHERICAL TRIANGLES.
Page 245: DIFFERENTIAL VARIATH-lNS OF SPHERIC
Page 249 and 250: APPROXIMATE SOLUTION OF SPHERICAL T
Page 251 and 252: APPROXIMATE SOLUTION OF SPHERICAL T
Page 253 and 254: MISCELLANEOUS PROBLEMS. Also, by th
Page 255 and 256: MISCELLANEOUS PROBLEMS. 249 173- Le
Page 257 and 258: Ther'fore, cos D cos R sin T Substi
Page 259 and 260: MISCELLANEOUS PROBLEMS. 253 18!^ TJ
Page 261 and 262: whence and similarly .MISCELLANEOUS
show all

31151029036484.pdf

Create successful ePaper yourself

Delete template?

Save as template?