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<br />TRIGONOMETRIC FORMULAE
<br />)S B B B
<br />>---I
<br />c c
<br />a a ...
<br />A b C AI b C A b C
<br />Right Triangle Oblique Triangles
<br />Solution of Right Triangles
<br />For Angle A. sin = a, cos - - , tan = b , cot = a, sec = b, cosec = a
<br />Given Required ,
<br />a,b A,B,c tanA=b=cotB,c= a= b==a IT. Q,
<br />a,c A,B,b sin A=a=cosB,b= (c+a)c-a)=c , a'
<br />A, a B, b, c
<br />A,b B,a,c
<br />A,c IB,a,b,i
<br />B - 90°-A, b = a cot A, c =sin A.
<br />B = 90°-A, a - b tan A, c =cos A.
<br />B - 907-A, a = c sin A. b = c cos A.
<br />Solution of Oblique Triangles
<br />Given 'Required a sin B
<br />A, B, a b, c, C b= sin A ' C = 180°- (A+B), c = sinin
<br />A, a, b B. c, C sin B C =180 - (A B), c = °sin C
<br />a sin A
<br />a, b, C A; 4, c A+B =180-C,tan} (A -B) _ (a -b) tan'} (A+B)
<br />•a+b
<br />a"sin C
<br />sin A
<br />a, b, c A, B, C s= a+z+c, sine A =A (s- b (s -c),
<br />sin } B = (s-a)cs-c), C=180°-(A•+B)
<br />a
<br />a, b, c Areas=°+Z+c, area= s(s-a) (s -b) (s -o
<br />A, b, c Area area= b c sin A
<br />2
<br />A,B,C,a Area area= a' sin B sin C
<br />2 sin A
<br />REDUCTION TO HORIZONTAL
<br />Horizontal distance -slope distance multiplied by the
<br />cosine of the vertical angle. Thus. for a slope distance oface
<br />403.6 1t. and a vertical angle of 4' 40' -the cosine of mope dsst x
<br />4. 40'. taken from a table of natural trigonometrical
<br />functions,..9967, and horizontal d4tance=403.6 x .9967
<br />•402.27 fl.
<br />Horimmal distance
<br />Horizontal distance aim -Slope distance minus slope dis-
<br />tance times (I - cosine of vertical angle). Using the same figures as in the preceding Example -
<br />Cos.4' 40' -.9967. 1-.9967..0033.403.6x.00.73.1.33 ft. Horizontal dist.. 403.6 -1.33- 402.27 ft.
<br />When the rise is known, the horizontal distance may be fuund by the following approximate
<br />rule: -the slope distance less the square of the rise divided by twice the dope distance. Thus,
<br />for a slope distance of 372.5 ft., and a rise of 15 It. the twimntal distance=
<br />372.5- 15X15
<br />2X372.5 =372.5-.30=372.211.
<br />10
<br />Table IX. -Natural Trigonometrical Functions
<br />Angle
<br />Sin
<br />'ran I
<br />Cot
<br />Cos,
<br />Angle
<br />Sin
<br />Tan
<br />Cot
<br />Cos
<br />0,
<br />0
<br />0
<br />0
<br />00
<br />1
<br />90
<br />O/
<br />8
<br />.1392
<br />.1405
<br />7.115
<br />.99027
<br />82
<br />10
<br />.0029
<br />.0029
<br />343.8
<br />I
<br />50
<br />10
<br />.1421
<br />.1435
<br />6.968
<br />.98986
<br />50
<br />20
<br />.0058
<br />.0058
<br />171.9
<br />.99998
<br />40
<br />20
<br />.1449
<br />.1465
<br />6.827
<br />.98944
<br />40
<br />30'
<br />.0087.0087
<br />114.6
<br />,99996
<br />30
<br />30
<br />.1478
<br />.1495
<br />6.691
<br />.98902
<br />30
<br />40
<br />.0116
<br />,0116
<br />85.94
<br />,99993
<br />20
<br />40
<br />,1507
<br />.1524
<br />6.561
<br />.98858
<br />20
<br />50
<br />.0145
<br />.0145
<br />68.75
<br />,99989
<br />10
<br />50
<br />.1536
<br />.1554
<br />6.435
<br />.98814
<br />10
<br />1
<br />.0175
<br />.0175
<br />57.29
<br />.99985
<br />89
<br />9
<br />.1564
<br />.1584
<br />6.314
<br />.98769
<br />81
<br />10
<br />.0204
<br />.0204
<br />49.10
<br />.99979
<br />50
<br />10
<br />.1593
<br />.1614
<br />6.197
<br />.98723
<br />50
<br />20
<br />.0233.0233
<br />42.96
<br />.99973
<br />40
<br />20
<br />.1622
<br />.1644
<br />6.084
<br />.98676
<br />40
<br />30
<br />.0262
<br />.0262
<br />38.19
<br />.99966
<br />30
<br />30
<br />.1650
<br />.1673
<br />5.976
<br />.98629
<br />30
<br />40
<br />.0291
<br />.0291
<br />34.37
<br />.99958
<br />20
<br />• 40
<br />.1679
<br />.1703
<br />5.871
<br />,98580
<br />20
<br />50
<br />.0320
<br />.0320
<br />31.24
<br />.99949
<br />10
<br />50
<br />.1708
<br />.1733
<br />5.769
<br />.98531
<br />10
<br />2
<br />.0349
<br />.0349
<br />28.64
<br />.99939
<br />88
<br />10 '
<br />.1736
<br />.1763
<br />5.671
<br />.98481
<br />80
<br />10
<br />.0378
<br />.0378
<br />26.43
<br />.99929
<br />50
<br />10
<br />.1765
<br />.1793
<br />5.576
<br />.98430
<br />50
<br />20
<br />.0407
<br />.0407
<br />24.54
<br />.99917
<br />40
<br />.20
<br />.1794
<br />.1823
<br />5.485
<br />.98378
<br />40
<br />30
<br />.0436
<br />.0437
<br />22.90
<br />.99905
<br />30
<br />30
<br />.1822
<br />.1853
<br />5.396
<br />.98325
<br />30
<br />40
<br />.0465
<br />.0466
<br />21.47
<br />.99892
<br />20
<br />40
<br />.1851
<br />.1883
<br />5.309
<br />.98272
<br />20
<br />50
<br />.0494
<br />.0495
<br />20.21•
<br />.99878
<br />10
<br />50
<br />.1880
<br />.1914
<br />5.226
<br />.98218
<br />10
<br />3 "
<br />,0523
<br />,0524
<br />19.08
<br />.99863.
<br />87
<br />11
<br />.1908
<br />.1944
<br />5.145
<br />.98163
<br />79
<br />10
<br />.0552
<br />.0553
<br />18.07
<br />.99847
<br />50
<br />10
<br />.1937
<br />.1974
<br />5.066
<br />.98107
<br />50
<br />20
<br />.0581
<br />.0582
<br />17.17
<br />,99831
<br />40
<br />20
<br />.1965
<br />.2004
<br />4.989
<br />.98050
<br />40
<br />30
<br />.0610
<br />.0612
<br />16.35
<br />,99813
<br />30
<br />30
<br />.1994
<br />.2035
<br />4.915
<br />.97992
<br />30
<br />40
<br />.0640
<br />:0641
<br />15.60
<br />,99795
<br />20
<br />40
<br />.2022
<br />.2065
<br />4.843
<br />.97934
<br />20
<br />50
<br />.0669
<br />.0670
<br />14.92
<br />,99776
<br />10
<br />50
<br />.2051
<br />.2095
<br />4.773
<br />.97875
<br />10
<br />4
<br />.0698
<br />.0699
<br />14.30
<br />.99756
<br />86
<br />12
<br />.2079
<br />.2126
<br />4.705
<br />.97815
<br />78
<br />10
<br />.0727
<br />.0729
<br />13.73
<br />.99736
<br />50
<br />10
<br />.2108
<br />.2156
<br />4.638
<br />.97754
<br />50
<br />20
<br />.0756
<br />.0758
<br />13.20
<br />.99714
<br />40
<br />20
<br />.2136
<br />.2186
<br />4.574
<br />.97692
<br />40
<br />30
<br />.0787
<br />12.71
<br />.99692
<br />30
<br />30
<br />.2154
<br />.2217
<br />4.511
<br />.97630
<br />30
<br />40
<br />.0785
<br />.0814
<br />.0816
<br />12.25
<br />.99668
<br />20
<br />40
<br />.2193
<br />.2247•
<br />4.449
<br />.97566
<br />20 -
<br />50
<br />.0843
<br />.0846
<br />11.83
<br />.99644
<br />10
<br />50
<br />.2221
<br />.2278
<br />4.390
<br />.97502
<br />10
<br />b
<br />.0872
<br />.0875
<br />11.43
<br />.99619
<br />85
<br />13
<br />.2250
<br />.2309
<br />4.331
<br />.97437
<br />77
<br />10
<br />.0901
<br />.0904
<br />11.06
<br />'.99594
<br />50
<br />10
<br />.2278
<br />.2339
<br />4.275
<br />.97371
<br />`�50
<br />20
<br />.0929
<br />.0934
<br />10.71
<br />.99567
<br />40
<br />20
<br />.2306
<br />.2370
<br />4.219
<br />.97304
<br />40-
<br />30
<br />.0958
<br />.0963
<br />10.39
<br />.99540
<br />30
<br />30
<br />.2334
<br />.2401
<br />4.165
<br />.97237
<br />30
<br />40
<br />.0987
<br />.0992
<br />10.08
<br />.99511
<br />20
<br />40
<br />.2363
<br />.2432
<br />4.113
<br />.97169
<br />20
<br />50
<br />.1016
<br />.1022
<br />9.788
<br />.99482
<br />10
<br />50
<br />.2391
<br />.2462
<br />4.061
<br />.97100
<br />10
<br />6
<br />.1045
<br />.1051
<br />9.514
<br />.99452
<br />84
<br />14
<br />.2419
<br />.2493
<br />4.011
<br />.97030
<br />76 -
<br />10
<br />.1074
<br />.1080
<br />9.255
<br />.99421
<br />50
<br />10
<br />.2447
<br />.2524
<br />3.962
<br />.96959
<br />50
<br />20
<br />.1103
<br />.1110
<br />9.010
<br />.99390
<br />40
<br />20
<br />.2476
<br />.2555
<br />3.914.
<br />.96687
<br />40
<br />30
<br />.1132
<br />.1139
<br />8.777
<br />.99357
<br />30
<br />30
<br />.2504
<br />.2586
<br />3.867
<br />,96815
<br />30
<br />40
<br />.1161
<br />:1169
<br />8.556
<br />.99324
<br />20
<br />40
<br />.2532
<br />.2617
<br />3.821
<br />.96742
<br />20
<br />50
<br />.1190
<br />.1198
<br />8.345
<br />.99290
<br />10
<br />50
<br />.2560
<br />.2648
<br />3.776
<br />.96667
<br />10
<br />7
<br />.1219
<br />.1228
<br />8.144
<br />.99255
<br />83
<br />15.2588
<br />.2679
<br />3.732
<br />.96593
<br />75
<br />10
<br />.1248
<br />.1257
<br />7.953
<br />.99219.
<br />50
<br />10
<br />,2616
<br />.2711
<br />3.689
<br />.96517
<br />50
<br />20
<br />.1276
<br />.1287
<br />7.770
<br />.99182
<br />40
<br />20
<br />.2644
<br />.2742
<br />3.647
<br />.96440
<br />40
<br />30
<br />.1305
<br />..1317
<br />7.596
<br />.99144
<br />30
<br />30
<br />.2672
<br />.2773
<br />3.606
<br />.96363
<br />30
<br />40
<br />.1334
<br />.1346
<br />7.429
<br />.99106
<br />20
<br />40
<br />.2700
<br />.2805
<br />3.566
<br />.96285
<br />20
<br />50
<br />.1363
<br />.1376
<br />7.269
<br />.99067
<br />50
<br />.2728
<br />.2836
<br />3.526
<br />.96206
<br />10
<br />B210
<br />74
<br />Cos
<br />Cot
<br />Tan
<br />Sin
<br />Angle
<br />Cos
<br />Cot
<br />Tan
<br />Sin
<br />Angle
<br />to
<br />41.
<br />11
<br />
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