|
TRIGONOMETRIC FORM ULrE
<br />B 8 B
<br />A�a As!� A�
<br />b C �b C b I
<br />Righ[ Triangle Oblique Tnangles -J
<br />Solution of Right Triangles
<br />For Angle A. sin = °�„ cos = �, tan = 6, cot = o, sec = b, cosec = °
<br />Given RwuirM
<br />a,b A, B,c tanA=6=cot B,r= a'}6'=0 l+Q,
<br />a,c A, B,b sinA=°=cos B,6= (c+o)c-a)=c 1-2
<br />A, a B, b, c B = 90°-A, 6 = a cot A, r = sin A.
<br />A, b B, a, c B = 90°-A, a = b tan A, c =cos A.
<br />A,c B, a,b 8=90°-A,a=rsin A,6=rcos A.
<br />Solution of ObliqueTrianglea
<br />Given RwurM
<br />A, B,a b, c,C 6=sinnA'C=180°-(A+B),c=s A
<br />A, a,b B, <,C sin B=bsianA C=180°-(A+B).c=snA
<br />a, b, C A, $, r A+B=180°-C,tan}(A-B)=(O-b) tan}(A+B)
<br />a}b
<br />a sin C
<br />a sin A
<br />a. b. a A, B. C s= a}2}c sin } A = (s-bD c -c)•
<br />sin } B = (s- n �s-0), C=180°-(A+B)
<br />a, b, r Area s=°+z+c, area= s(s-d) (s -b) (s -c)
<br />A, b, c Area area= b r sin A
<br />2
<br />A, B, C,a Area area= °� sin B sin C
<br />2 sin A
<br />REDUCTION TO HORIZONTAL
<br />Horiwnul tliaunre=slope tliauntt mulliDliM DY the
<br />e of Ne v<rtiul angle. Thus, lora slope tliaunre of dat9pce g
<br />d03.6 Il. and a verliral angle of 4° 40'-l6e ws n< 04 Sdotl=
<br />° 40'. uMen Iwm a ubk of Kral ni m nul - Vers, Pp61a
<br />lunczionz.=,996], and 6omm�ul diaunre=igi.6 xe199fi]
<br />X40['3] Il' Horimnul aisunm
<br />H°rimnul disunre aln=Slope distanre minut slope tlis�.
<br />es (1 -cosine of vertid angle). U[ir,g abe vme figures a[ in a6e preceding examDle-
<br />Coa d°40'=.9961.1 -9961=.0013.403.6x.0033=1 &1 R. Horimnuldist.=IN.6-1.33=dOt 2] ([.
<br />isu n° al disu y b I untl by4g` follow
<br />rvl<w f tM1 u tl�vidM by t he al 1X �d6sunc�xT6uP
<br />for a slopezd'sunre[ol 3]2.5 Iz�M1andaa of lS (l� be M1°r nul aisz =
<br />ns I5X 15 l
<br />3]2.5-2X3T2.5 -3)2.5-.30=3R.211.e
<br />l�
<br />4
<br />Y
<br />Table IX. -Natural Trigonometrical Functions.
<br />Angle
<br />Sin
<br />7'an
<br />Cot
<br />Coe
<br />Angle
<br />Sin
<br />Tan
<br />Cot
<br />Coe
<br />0
<br />10
<br />0
<br />0
<br />00
<br />343.8
<br />1
<br />I
<br />90
<br />50
<br />8
<br />.1392
<br />.1405
<br />7.115
<br />.99027
<br />82
<br />20
<br />,0029
<br />.0058
<br />.0029
<br />.0058
<br />171.9
<br />40
<br />10
<br />20
<br />.1421
<br />.1435
<br />6.968
<br />.98986
<br />50
<br />30
<br />.0087
<br />.0087
<br />114.6
<br />.99998
<br />.99996
<br />30
<br />30
<br />.1449
<br />.1478
<br />.1465
<br />6.827
<br />6.691
<br />.98944
<br />40
<br />30
<br />40
<br />.0116
<br />.0116
<br />85.94
<br />.99993
<br />20
<br />40
<br />.1507
<br />.1995
<br />6.561
<br />.98902
<br />20
<br />50
<br />.0145
<br />.0145
<br />68.75
<br />.99989
<br />10
<br />50
<br />.1536
<br />.1524
<br />.1554
<br />6.435
<br />.98858
<br />.98814
<br />]0
<br />1
<br />.0175
<br />.0175
<br />57.29
<br />.99985
<br />80
<br />9
<br />.1564
<br />6.314
<br />81
<br />10
<br />.0204
<br />.0204
<br />49.10
<br />.99979
<br />50
<br />10
<br />.1593
<br />.1584
<br />.1614
<br />6.197
<br />.98769
<br />50
<br />20
<br />.0233
<br />.0233
<br />42.96
<br />.99973
<br />40
<br />20
<br />.1622
<br />6.084
<br />.98723
<br />40
<br />30
<br />.0262
<br />.0262
<br />38.19
<br />.99966
<br />30
<br />30
<br />.1650
<br />.1644
<br />.1673
<br />5.976
<br />.98676
<br />30
<br />40
<br />.0291
<br />.0291
<br />34.37
<br />.99958
<br />20
<br />40
<br />.1679
<br />5.871
<br />.98629
<br />20
<br />50
<br />.0320
<br />.0320
<br />31.24
<br />.99949
<br />10
<br />50
<br />.1708
<br />.1703
<br />.1733
<br />5.769
<br />.98$80
<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 />80
<br />10
<br />.0378
<br />.0378
<br />26.43
<br />.99929
<br />50
<br />10
<br />.1765
<br />.1793
<br />5.576
<br />.98481
<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 />.98430
<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 />.98378
<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 />.98325
<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 />.98272
<br />.98218
<br />10
<br />S
<br />.0523
<br />.0524
<br />19.08
<br />.99863
<br />87
<br />11
<br />.1908
<br />.1944
<br />5.145
<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 />.98163
<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 />.98107
<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 />.98050
<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 />.97992
<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 />.97934
<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 />78
<br />]0
<br />.0727
<br />.0729
<br />13.73
<br />.99736
<br />50
<br />10
<br />.2108
<br />.2156
<br />4.638
<br />.97815
<br />50
<br />20
<br />.0756
<br />.0758
<br />13.20
<br />.997!4
<br />40
<br />20
<br />.2136
<br />.2186
<br />9.574
<br />.97754
<br />40
<br />30
<br />.0785
<br />.0787
<br />12.71
<br />.99692
<br />30
<br />30
<br />.2164
<br />.2217
<br />4.511
<br />.97692
<br />30
<br />40
<br />.0814
<br />.0816
<br />12.25
<br />.99668
<br />20
<br />40
<br />.2193
<br />.2247
<br />4.449
<br />.97630
<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 />.97566
<br />.97502
<br />10
<br />5
<br />.0872
<br />.0875
<br />11.43
<br />.99619
<br />85
<br />13
<br />.2250
<br />.2303
<br />4.331
<br />77
<br />]0
<br />.0901
<br />.0904
<br />11.06
<br />.99594
<br />50
<br />10
<br />.2278
<br />.2339
<br />4.275
<br />.97437
<br />50
<br />20
<br />.0929
<br />.0934
<br />10.71
<br />.99567
<br />40
<br />20
<br />.2306
<br />.2370
<br />9.219
<br />.97371
<br />40
<br />30
<br />.095P
<br />.0963
<br />10.39
<br />,99540
<br />30
<br />30
<br />.2334
<br />.2401
<br />4.165
<br />.97304
<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 />.97237
<br />20
<br />50
<br />.1016
<br />.1022
<br />9.788
<br />.99482
<br />]0
<br />50
<br />.2391
<br />.2462
<br />4.061
<br />.97169
<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 />76
<br />10
<br />.1074
<br />.1080
<br />9.255
<br />.99421
<br />50
<br />10
<br />.2447
<br />.2524
<br />3.962
<br />.97030
<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 />40
<br />30
<br />.1132
<br />.1139
<br />8.777
<br />.99357
<br />30
<br />30
<br />.2504
<br />.2586
<br />3.867
<br />.96887
<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 />20
<br />50
<br />.1190
<br />.1198
<br />8.345
<br />.99290
<br />10
<br />50
<br />.2560
<br />,2648
<br />3.776
<br />.96742
<br />.96667
<br />10
<br />7
<br />.1219
<br />.1228
<br />8.144
<br />.99255
<br />8$
<br />15
<br />.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 />50
<br />20
<br />.1276
<br />.1287
<br />7.770
<br />.99182
<br />40
<br />20
<br />.2644
<br />.2742
<br />3.647
<br />.96517
<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 />20
<br />50
<br />.1363
<br />.1376
<br />7.269
<br />.99067
<br />]0
<br />50
<br />.2728
<br />.2836
<br />3.526
<br />.96285
<br />.96206
<br />10
<br />82
<br />74
<br />Cos
<br />Cot
<br />Tan
<br />Sin
<br />Angle
<br />Coa
<br />Cot
<br />Tan
<br />Sin
<br />Angle
<br />
|