in each bent are 10 feet apart; bents are identified byletters; and piles, by numbers. The distance betweenadjacent transit setups in the base line is 10/sin 84°, or10.05 feet.Bents are located 20 feet apart. The distancefrom the center-line base line transit setup at station7 + 41.05 to pile No. 3 is 70 feet. The distance fromstation 7 + 51.10 to pile No. 2 is 70 + 10 tan 6°, or70 + 1.05, or 71.05 feet. The distance from station7 + 61.15 to pile No. 1 is 71.05 + 1.05, or 72.10 feet.The distance from station 7 + 31.00 to pile No. 4 is70 - 1.05, or 68.95 feet; and from station 7 + 20.95 topile No. 5 is 68.95 – 1.05, or 67.90 feet.You can determine the angle you turn, at a controlstation, from the base line to any pile location bytriangle solution. Consider pile No. 61, for example.This pile is located 240 + 72.10, or 312.10 feet, fromstation 7 + 61.15 on the base line. Station 7 + 61.15 islocated 1,038.83 – 761.15, or 277.68 feet, from controlstation 10 + 38.83. The angle between the line fromstation 7 + 61.15 through pile No. 61 and the base linemeasures 180°- 84°, or 96°. Therefore, you are dealingwith the triangle ABC shown in figure 10-31. You wantto know the size of angle A. First solve for b by the lawof cosines, in which b^{2} = a^{2 }+ c^{2}- 2ac cos B, as follows:b^{2 }= 312.102 + 277.68^{2}- 2(312.10)(277.68) cos 96°b = 438.89 feetKnowing the length of b, you can now determinethe size of angle A by the law of sines. Sin A = 312.10sin 96°/438.89, or 0.70722. This means that angle Ameasures, to the nearest minutes, 45°00´.Figure 10-31.—Trigonometric solution for pile No. 61.Figure 10-32.—Trigonometric solution for pile No. 65.To determine the direction of this pile fromcontrol station 4 + 43.27, you would solve the triangleDBC shown in figure 10-31. You do this in the samemanner as described above. First solve for b using thelaw of cosines and then solve for angle D using thelaw of sines. After doing this, you find that angle Dequals 47°26´.It would probably be necessary to locate in thisfashion only the two outside piles in each bent; thepiles between these two could be located by measuringoff the prescribed spacing on a tape stretched be-tween the two. For the direction from control station10 + 38.83 to pile No. 65 (the other outside pile inbent M), you would solve the triangle shown in figure10-32. Again, you solve for b using the law of cosinesand then use the law of sines to solve for angle A.For each control station, a pile location sheet likethe one shown in figure 10-33 would be made up. Ifdesired, the direction angles for the piles between No.61 and No. 65 could be computed and inserted in theintervening spaces.Figure 10-33.—File location sheet.10-29