: rad

To find the angle A.
AC 586...

207767898

10.000000 : : A D 327.11

2:514693
: cos A 56° 4' 5"

9.746795
To find the angle B.
As B C 649

2.812245
: rad ..

10.000000 ::D B 429.89

2.633356 : cos B 48° 31' 5"

9.8211ll And L AC B is the supplement of A + B, or 75° 24' 50%

Second Method of Solution. All the angles are acute, because the sum of the squares of the two less sides exceed the square of the greatest,

A B 757 log 2:879096
B C 649 log 2.812245
AC 586 log 2-767898
2)1992 20.602060 constant logarithm

996 log 2.998259
239 log 2-378398
347 log 2.540329
410 log 2:612784

2)31•131830

15.565915 log A B + log BC 5.691341 diff. 9•874574 log sin 48° 31' 3" ZB log A B + log AC 56646994 diff. 9.918921 log sin 56 4.4 LA log B C + log AC 5.580143 diff. 9.985772 log sin 75 24 49 LC

Third Method of Solution.
A B 757 log 2.879096
B C 649 log 2:812245
A C 586

5.691341
2) 1992

996

239 log 2:378398

347 log 2:540329 constant log 20

24:918727

2)19:227386
24° 15' 32' sin 9•613693

2
48 31 4 B

EXAMPLES FOR EXERCISE. Let ABC (see first figure p. 116) represent any oblique angled triangle.

1. Given A B 697, 2 A 81° 30' 10", and 6 B 40° 30' 44", to find the other parts ?

Answer, AC 534, B C 813, and 2 C 57° 59' 4". 2. If A C = 720·8, A = 70° 5' 22", and B = 59° 35' 36", required the other parts?

Answer, A B 643.2, B C 785.8, and 4C 50° 19'6". 3. Given B C 980'1, A 7° 26' 26", and < B 106° 2' 23', to find the other parts ?

Answer, A B 7284, AC 7613:3, and C 66° 51' 11". 4. Given A B 896.2, B C 328:4, and EC 113° 45' 20', to find the other parts ? Answer, AC 712, 2 A 19° 35' 48", and Z B 46°38' 52".

5. Given A C 4627, B C 5169, and 4 A 70° 25' 12'', to find the other parts ?

Answer, A B 4328, 2 B 57° 29' 58", and 2 C 52° 4' 52". 6. Given A B 6.216, BC 7.853, and 4 A 70° 34' 40", to find the other parts?

Answer, A C 6319, B 51° 47' 48", and 2 C 50° 37' 30'. 7. Given A C 627, A B 430, and 2 C 42° 53' 38", to find the other

parts?

Answer, 4 A 54° 8' 22", or 40° 4' 18", B 82° 57' 56", or 97° 2' 4", and B C 512, or 406•7.

8. Given A B 718, B C 629, and 4 A 29° 52' 34", to find the other parts?

Answer, 2 C 34° 39' 11", or 145° 20' 49", 2 B 115° 28' 15", or 4° 46' 37", and A C 1140, or 105·1.

9. Given A C 28:48, B C 71.34, and < B 23° 20' 58", to find the other parts?

Answer, 4 A 96° 53' 33", or 83° 6' 27", LC 59° 45' 291, or 73° 32' 35", and A B 62.08, or 68:91.

10. Given A C 484:2, A B 9684, and 4 A 75° 31' 21", to find the other parts ?

Answer, B C 968:4, 2 B 28° 57' 18", and LC 75° 31' 21". 11. Given A B 1234:5, B C 620:8, and . B 138° 39' 8", to find the other parts?

Answer, A C 1749:3, A 13° 33' 34”, and 2 C 27° 47' 18". 12. Given A C 72:48, B C 60·2, and 2 C 31° 1' 10%, to find the other parts? Answer, A B 37•4, A 56° 2' 45', and Z B 92° 56' 5''. 13. Given A B 912-4, B C 63967, and A C 428:5, to find the angles ?

Answer, 2 A 39° 5' 36"', B 24° 59' 8", and < C 115° 55' 16''. 14. Given A B 793.8, B C 481:6, and A C 5000, to find the angles ? Answer, 2 A 35° 15' 32", B 36° 49' 18", and 2 C 107° 55' 10'.

15. Given A B 100:3, BC 100:3, and AC 100-3, to find the angles ?

Answer, 2 A 60°, 2 B 60°, and 2 C 60°.

parts?

parts ?

16. Given A B 92:6, B C 463, and A C 71.2, to find the angles ? Answer, LA 29° 17' 22", Z B 48° 47' 31", and 2 C 101° 55'8''. 17. Given A B 4963, B C 5124, and A C 5621, to find the angles ? Answer, 2 A 57° 30' 28'', 2 B 67° 42' 36', and 2 C 54° 46' 56''. 18. Given A B 728:1, B C 6147, and AC 5838, to find the angles ? Answer, 2 A 54° 32' 52", 2 B 50° 40 58", and 2 C 74° 46' 10".

19. Given A B 96-74, B C 83:29, and AC 111:42, to find the angles ?

Answer, 2 A 46° 30 45”, Z B 176° 3' 45', and < C 59° 25' 30". 20. Given A B 363•4, B C 148:4, and 4 B 102° 18' 27", to find the other parts ?

Answer, 2 A 20° 9' 19", 2 B 102° 18' 27", and 2 C 57° 32' 16". 21. Given A B 632, B'C 494, and 2 A 20° 16', to find the other parts, C being acute?

Answer, LC 26° 18' 19", 2 B 133° 25' 41", and AC 1035.86. 22. Given A B 53.9, A C 46° 21', and 2 B 58:16, to find the other

Answer, 2 A 38° 58', _ C 82° 46', and B C 34:16. 23. Given A B 2163, B C 1672, and 4 C 112° 18' 22", to find the other parts ?

Answer, AC 877-2, 2 B 22° 2' 16", and 2 A 45° 39' 22". 24. Given A B 496, B C 496, and 2 B 38° 16', to find the other

Answer, AC 325:1, 2 A 70° 52', and 2 C 70° 52'. 25. Given A B 428, Z C 49° 16', and AC + BC 918, to find the other parts, 4 B being obtuse ? 1

Answer 2 A 33° 44' 48", 2 B 91° 59' 12", AC 564 49, and BC 353.5.

26. Given A C 126, 6 B 29° 46', and AB B C 43, to find the other parts ?

Answer, 4 A 55° 51' 32", _ C 94° 22' 28", A B 253-54, and BC 210-54.

27. Given A B 1269, A C 1837, and 2 A 53° 16' 20", to find the other parts ?

Answer, 2 B 83° 23' 47", Z C 43° 19' 53", and BC 1482'16. 28. Given A B 821:9, AC 640-3, and 4 A 80° 24', to find the other parts ?

Answer, 1 B 41° 26' 18", LC 58° 9' 42'', and B C 953.915. 29. Given A B 67'4, B C 53:1), and 2 B 93° 26' 44", to find the other parts ?

Answer, 2 A 36° 54' 23', LC 49° 38' 53", and A C 88282, 30. Given A C 29674, BC 31283, and 2 C 121° 5' 38", to find the other parts ?

Answer, 2 A 30° 18' 25", B 28° 35' 57", and A B 53084:5. 31. Given A B 73, AC 100, and A 2° 14' 31", to find the other parts ?

APPLICATION OF THE PRINCIPLES OF TRIGONOMETRY TO THE DETERMINATION OF THE HEIGHTS AND DISTANCES OF REMOTE OR INACCESSIBLE OBJECTS.

In this useful application of trigonometry, a base line is always supposed to be measured, or given in length; and by means of a quadrant, sextant, circle, theodolite, or some other instrument for measuring angles, such angles are measured as connected with the base line, and the objects whose heights or distances it is proposed to determine, enable us to compute, from the principles of trigonometry, what those heights or distances are.

Sometimes, particularly in marine surveying, horizontal angles are determined by the compass; but the varying effect of surrounding bodies on the needle, even in situations little removed from each other, and the general construction of the instrument itself, render it unfit to be applied in the determination of angles where any thing like precision is required.

The following examples present sufficient variety to guide the student in determining what will be the most eligible mode of proceeding in any case that is likely to occur in practice.

EXAMPLE I. Wanting to know the distance of an inaccessible object C, (see first figure, p. 116) I measured a base A B of 486 yards. At A, I found the angle C A B subtended by the object, and the other end of the line, to be 88° 12'; and at B the angle C B A was observed to be 54° 48'; required the distance of the object from each of the stations A and B ?

The sum of the angles A and B is 143°, which, taken from 180°, leave 37° for the angle C. Hence as sin 2 C 376

9.779463 : A B 486 yards

2.686636 :: sin 2 A 88° 12'..

9.999786

12-686422 :B C 807•2 yards

2.906959

EXAMPLE II. Being desirous of finding the distance between two distant objects, C and D, I measured a base A B of 384 yards, on the same horizontal plane, with the objects C and D. At A, I found the angle D AB=