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<br />1 XII -TRIGONOMETRIC FORMULAE
<br />Natural Trigonometrical Functions
<br />B
<br />Angle. Sin: Tan. Sec. Cosec. Cotg. Cosin. Angle.Sin. Tan. Sec. Cosec. Catg. Cosin. J a C a c a
<br />of i .o"i I of
<br />''LII 32 .5399 .02491.17921.8871.600.84505 58 39 C20ff:S098L23631.SS91.235.7in551 C dA C
<br />_
<br />10 .5324 .6299 1.1813 1.878 1.590 .84650 50 10.6316.8146i1.23981.5S3I 1.228.77531 .50 I -!I
<br />'Right Triangle Oblique Triangles
<br />�l 20 .5343 .6330 1.19-35 1.870 1.50 .84495 -'40 20 .6335 .81951,20391.57 1.220 -77347 40 +
<br />i ,1 30 .5373 .6371 1.1557 1.861 1.576 :34339 30 {'30 :6361 ;82431_29591.57 1.213 .77162 30 Il 1 Solution of- Right Triangles
<br />1, 4 .5308 .6412 t. 1879 1.853 1 . 660 .84182 20 40.6383,82921.29911.5671."206.769771 20 " 1;! a b a b C c
<br />LFor Angle A. sin = - , cos = , tan= cot = sec, = cosec =
<br />50 .5422 .6453 I ,1901 1.844 1.550 .84025 40. 50 .6406 •834'21-. 30321.561 1,199 , 7f;791' 10 g C C b (L f? a
<br />33 .544G -6.4941 .1924 1.836 1,540 ,83867 57 40. :6428 .8391 1:30541.5:;61.193 .76;604 50 'i' Given Required (G 2_
<br />10 .5471 .6536 t. 1946 1.82 1.530 .53708 50 - 10 .6450 '.5441 1.30861.5501.185 .76417 50 a, b _q B , C . tan. S = - - cot B, C = 1 a= i b= = a 1 } (1.2
<br />20 .5495 .65771 .10691.8201.520.83549 40.. 20 .647'1.:84911.31181.5451.173 .76229 40 b v.
<br />a2
<br />30 .5519 .0619 1 .1992 1.812 1,511 -83359 30 6494 30 . .85411.3151 1.540 1.171 .76041 30 a
<br />40 ,5544 .66611 , 2015 1 . 804 1-501 ,53228 20 40 ,6517 .85911.3184 1-535 1.164 ,75851 20 a, c A, B, b . sin :4 = C =cos B. b = \/ (c } u) (c -a) = C 1 = 02
<br />1 50 .5563 .67031, 2039 1:7961 492 .83066 10 50 . G530 .8642 1, 3217 1.529 1.157 .75661 10 . r a xx
<br />34 .5592 .6745 1.206 1.7 561 1.453.82904 66 41 .6,EG931.32511.5241.150:754i149 A,'a' B, b, C B=90o-A, b = a cot.3, c= sin A. Y'
<br />10 .5610 .6787 1.2086 1.7811.473 .82741 50 10 16533 .8744 1.32&4 1.519 1.144 .75250 50
<br />} ' n 20 .5640 .68ZO 1.2110 1.773 1.464 .52577 40 20 .6404 ,57961.33181.5141.137 .75638 40 1 0
<br />_ ; b B a, c B=90 -�,a , Utan 4,c=
<br />30 .5664 :6373 0.2134 1.7661.455 :52413 30. 30 : GG26 : &S471.3352 1.509 1.130 .74596 30 cos A
<br />40 •�.5683 .6916 1.2158 1..7581.446 :82243 '20 40 .6648 .8899 1.3386 1.504 1.124 .,4703 20 :1, r, B a; b B=900 -
<br />d, 2 = csin:l,'b= ccos,A, 1
<br />50 .5712 .6959 1.2183 1.7511:437 .82082 10 50 ,6670 .8952 1.3421 1.40D 1.117 .74509 10
<br />Solution of Oblique Triangles
<br />35 .5736 ; 7002 1.2208 1.743 1.428 .81915 55 42 .6691 .9004 1.3456 1.494 1. ill .74314 48 - �'• k ,
<br />10 .5760 .7046 1-2233 1.736 1.419 .51748 50 10 .6713 .9057 1.3492 1.490 1.104 . 74120 50 Gii en Required I 'a sin B a sin C
<br />il+ � I ',_ ��-B).c=
<br />!• 1 20 .5783 .7011.2258 f.720 1.411 ,815580 40 20 ,6734 .0110 1.3527 1.485 1.098 :73924 40 A, B, a b, c, G 4 sin 2 sin A
<br />30 .6807 .713.3 1.2283 1.722 1.402 .81412 30 30 .6756 0163 J.3563 1.4801.001 .73725 30 b sin A asin'C 1
<br />40 .5831 .7177 1 .2309 1.715 1.393 .51242 20 40 ,6777 .9217 1.3600 1.476 1,085 :73531 20 -Aa b B, c, C sin 7 = a C = 180°-(-4 B), c = sin A
<br />50.5854,72211,23351.7081.385.51072 10 50 .6799 .92711.3630 1.471 1.070 .7:f333 10
<br />(a -h) tan .'_. (A+B) `
<br />36 .5873 .7265 1.2361 1,701 1.37G.80902 54 43 .6820 .9325 1.3673 1.466 1.072 .7313547 bC - A, B; o _ +B-1800 - G', tan ! (_i -B)=
<br />i .•.� 10 .5901 .73101.23371:8951.363 .80730 .501 10 .6841. .93501.3; 11 1..4621.066 .72937 50 a, , a + b
<br />• j'
<br />20 .5925 .7355 1.2413 1.688 1. S60 . 80558 40 20.6862:94351.37451.4571.00.72737 40 c _ (Ain C t1
<br />(, 30 -5948 .7400 1.2440 1.6S1 1.351 :80366 30 30 "6SS4 .9490 1.37SG 1.453 1.054 .72537 30 sin A E
<br />40 .5972 .7445 1.24G6 1.61-5 1.343 .50212 20 40.6905 .9545 1.3S24 1.448 1.04S -72337 .�20 a } b+c t �, h-N,•''c)
<br />'t 50.5995.74901.249'41.6651.335:£0038 10 50.6926.'00011.28631.4441.042.72136 .10 ,
<br />+, a, b, e �trB,C s= 2 ,sin_ -1=1 be • 1
<br />37 .6013 .7536 1.2521 1.662 1.327 ."r9S64 53' 44 � .6947 _9657 1.3902 1.g4C 1.636.71934 46 ,
<br />10 .6041 .7581 1.2549 1.655 1.319 .79655 50 to .6967 .9713 1.39411.4351.030 .71732 50 ( a)(s-e) 1
<br />20 .6065 .7627 1.2577 1.649 1.311 .79312 40 20.69S3.07701.39S01.4311.024.71529 40 sin:'-. B= C=180° -(/i+
<br />Jl
<br />1I+ a0
<br />30 .6053 .7673 1.2605 1.643 1 .303 . 79335 • 30 30 .7009 .9527 1.4020 1.427 1.018 .71325 30 a b 40.6111 .77201,26331.6361.295.79158 20 40.7030.98541.40611.4221:012-71121 •20 ' + + /I
<br />a arca
<br />Area
<br />2
<br />50 .6134 .776 1.2651 1. No 1.288 .789S0 10 50 .7050 .0942 1.4101 1.418 1.006 .70916 10
<br />• )� 38 .6157 .7813 1.2690 1.624 I.2S0 .78801 52 .7071 1. 1.414 1.414 1.. .70711 45 b c sin A
<br />(� �1 10 .G180 .786 1-27191.6181.272 .7S622 50 A, b, c - Area area = 2
<br />20 .6202 -790 1.2748 1.612 1,265 .78442 40
<br />30 .6223 .7954 1.277S 1.606 1'.257 .78261 30" a sin B sin C
<br />40.(1249.50021.)o081.G01I.2-0.75079 20 '; i1, B, C,a Area area = 2 sin A
<br />So .sen .305o1.2sa3I1.595 1.2421n 10 _ i REDUCTION TO HORIZONTAL
<br />! Horizontal distance=Slope distance multiplied by the
<br />cosine of the vertical angle. Thus: slope distance =319.4 ft.
<br />Cosin. Calg. Cosec.. Sec. Tan. Sin. Angle Cssin. Colg. Cosec. Seo. Tan. Sin, Angle ee Vert. angle=50 10'. From Table, Page I%. cos 5010'= ,
<br />w - j a•5� 9959. Horizontal distance=319.4X.J<J59=31&09 ft.
<br />$\oDe n�\e A horizontal distance also=Slope distance minus slope
<br />1 � Ve • A distance times (1 -cosine of vertical angle):' With the
<br />!.!!q same figures as in the preceding example, the follow
<br />.9959. 'J ing result is obtained- Cosine 5°10'=.9959.1-.0950=.00k1.
<br />Horizontal distance
<br />:." 319.4X.0041=1.31. 319.4-1.31=318.09 ft.
<br />When the rise is known, the horizontal distance is approximately: -the slope
<br />dist-ante less the square of the else divided by twice the slope distance. Thus: rise=14.%,
<br />slope distance=302.6 ft. Horizontal distance=302.6-2X 362.6 602.6-532-302.28 ft
<br />MADE IN V. 9. A.
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