For these problems I use Bowditch Table 7 in the green book, hopefully I can explain this so people can make sense of it. I've used this method about 60 times running up and down the coast and it comes out good. I use the form thats on my link.
Distance of an object by two bearings - A running fix can be obtained by utilizing a mathematical relationships involved. A ship steams past a landmark. At any point a bearing of a LIGHT is observed and expressed as degrees right or left of the course (a relative bearing if the ship is on course). At some later time, at a second bearing of the LIGHT is observed and expressed as before when the LIGHT is broad on the beam. The various triangles could be solved by trigonometry to find the distance from at any bearing. Distance and bearing provide a fix.
Example: A ship is steaming on course 315° speed 12 knots. At 1230 a lighthouse bears 340° and at 1300 it bears 000°
(1) Distance from the light at 1300 (2nd Brg).
(2) Distance form the light when it is broad on the starboard beam.
Table 7 (Green - Bowditch) or Table 18 (Red - Bowditch) provides a quick and easy solution. The table is entered with the difference between the course and first bearing along the top of the table, and the difference between the course and 2nd bearing at the left of the table. For each pair of angles listed, two numbers are given:
To find the distance from the LIGHT at the time of the 2nd bearing multiply the distance run between bearings by the first number in the left side of each column from Table 7.
To find the distance when the LIGHT is abeam, multiply the distance run between by the second number right side of each column from the table.
2nd bearing
1.19
1.09
1.42
1.28
_____
4.98 divided by 4 = 1.245
Abeam
0.83
0.78
0.98
0.92
______
3.51 divided by 4 = .8775
1st Brg
Time 1230
Bearing 340
Course 315
Diff 25
2nd Brg
Time 1300
Bearing 000
Course 315
Diff 45
Time x Speed = Distance Run
30m x 12 kts = 6.0 miles x 1.245 = 7.47 miles off 2nd Brg
Distance Run 6.0 miles x .8775 = 5.265 miles Abeam
