For piles located farther offshore, the triangulation method of location is preferred. A pile location diagram is shown in figure 10-30. It is presumed that the piles in section X will be located by the method just described, while those in section Y will be located by triangulation from the two control stations shown.

The base line measures (1,038.83 433.27), or 595.56 feet, from control station to control station. The middle line of piles runs from station 7 + 41.05, making an angle of 84 with the base line. The piles

in each bent are 10 feet apart; bents are identified by letters; and piles, by numbers. The distance between adjacent transit setups in the base line is 10/sin 84, or 10.05 feet.

You can determine the angle you turn, at a control station, from the base line to any pile location by triangle solution. Consider pile No. 61, for example. This pile is located 240 + 72.10, or 312.10 feet, from station 7 + 61.15 on the base line. Station 7 + 61.15 is located 1,038.83 761.15, or 277.68 feet, from control station 10 + 38.83. The angle between the line from station 7 + 61.15 through pile No. 61 and the base line measures 180- 84, or 96. Therefore, you are dealing with the triangle ABC shown in figure 10-31. You want to know the size of angle A. First solve for b by the law of cosines, in which b² = a² + c² - 2ac cos B, as follows:

b² = 312.102 + 277.68² - 2(312.10)(277.68) cos 96 

b = 438.89 feet

Knowing the length of b, you can now determine the size of angle A by the law of sines. Sin A = 312.10 sin 96/438.89, or 0.70722. This means that angle A measures, to the nearest minutes, 4500.

To determine the direction of this pile from control station 4 + 43.27, you would solve the triangle DBC shown in figure 10-31. You do this in the same manner as described above. First solve for b using the law of cosines and then solve for angle D using the law of sines. After doing this, you find that angle D equals 4726.

It would probably be necessary to locate in this fashion only the two outside piles in each bent; the piles between these two could be located by measuring off the prescribed spacing on a tape stretched be-tween the two. For the direction from control station 10 + 38.83 to pile No. 65 (the other outside pile in bent M), you would solve the triangle shown in figure 10-32. Again, you solve for b using the law of cosines and then use the law of sines to solve for angle A. For each control station, a pile location sheet like the one shown in figure 10-33 would be made up. If desired, the direction angles for the piles between No. 61 and No. 65 could be computed and inserted in the intervening spaces.