And A C 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.

15.565915

LA

24 49

4 C

5-691341 diff. 9.874574 log sin 48° 31' 3" B
5.646994 diff. 9'918921 log sin 56 4 4
5.580143 diff. 9.985772 log sin 75
Third Method of Solution.

A B 757 log 2.879096
B C 649 log 2.812245

EXAMPLES FOR EXERCISE.

Let A B C (see first figure p. 116) represent any oblique angled

triangle.

1. Given A B 697, ▲ A 81° 30′ 10′′, and 4 B 40° 30′ 44′′, to find the other parts?

Answer, A C 534, B C 813, and ▲ 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 4 C 50° 19' 6". 3. Given B C 980-1, 4 A 7° 26' 26", and 4 B 106° 2′ 23′′, to find the other parts?

Answer, A B 7284, A C 7613.3, and 4 C 66° 51′ 11′′. 4. Given A B 896'2, B C 3284, and C 113° 45′ 20′′, to find the other parts? Answer, AC 712, ▲ A 19° 35′ 48′′, and 4 B46°38′52′′. 5. Given A C 4627, B C 5169, and ▲ A 70° 25′ 12′′, to find the other parts?

Answer, A B 4328, 4 B 57° 29′ 58′′, and C 52° 4′ 52". 6. Given A B 6'216, B C 7.853, and 4 A 77° 34′ 40′′, to find the other parts?

Answer, A C 6319, 4 B 51° 47′ 48′′, and 4 C 50° 37′ 30′′. 7. Given A C 627, A B 430, and ▲ C 42° 53′ 38′′, to find the other parts?

Answer, 4 A 54° 8′ 22′′, or 40° 4′ 18′′, 4 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, C 34° 39′ 11′′, or 145° 20′ 49′′, ▲ 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′′, ▲ C 59° 45′ 29′′, or 73° 32′ 35′′, and A B 62.08, or 68.91.

10. Given A C 484·2, A B 968'4, and ▲ A 75° 31′ 21′′, to find the other parts?

Answer, B C 968 4, 4 B 28° 57′ 18", and 4 C 75° 31′ 21′′. 11. Given A B 12345, B C 6208, and B 138° 39′ 8′′, to find the other parts?

Answer, A C 1749-3, ▲ A 13° 33′ 34′′, and ▲ C 27° 47′ 18′′. 12. Given A C 72·48, B C 60·2, and 4 C 31° 1' 10", to find the other parts? Answer, A B 374, ▲ A 56° 2′ 45′′, and 4 B 92° 56′ 5′′. 13. Given A B 912.4, B C 639'7, and A C 428'5, to find the angles? Answer, A 39o 5′ 36′′, ▲ B 24° 59′ 8′′, and C 115° 55′ 16′′. 14. Given A B 793.8, B C 481-6, and A C 500'0, to find the angles ? Answer, 4 A 35° 15′ 32′′, 4 B 36° 49′ 18′′, and ▲ C 107° 55′ 10′′. 15. Given A B 100.3, BC 1003, and AC 1003, to find the angles? Answer, A 60°, ▲ B 60°, and 4 C 60°.

16. Given A B 92.6, B C 46·3, and A C 71.2, to find the angles ? Answer, 4 A 29° 17′ 22′′, ▲ B 48° 47′ 31′′, and ▲ C 101° 55′ 8′′. 17. Given A B 4963, B C 5124, and A C 5621, to find the angles? Answer, A 57° 30′ 28′′, ▲ B 67° 42′ 36′′, and C 54° 46′ 56′′. 18. Given A B 728 1, B C 614-7, and A C 583-8, to find the angles? Answer, A 54° 32′52′′, ▲ B 50° 40′ 58′′, and ≤ C74° 46′ 10′′. 19. Given A B 96 74, BC 83 29, and AC 11142, to find the angles?

Answer, A 46° 30′ 45′′, ▲ B 76° 3′ 45′′, and ≤ C 57° 25′30′′. 20. Given A B 363'4, B C 148'4, and 4 B 102° 18′27′′, to find the other parts?

Answer, ZA 20° 9′ 17′′, ▲ B 102° 18′ 27′′, and ▲ C 57° 32′ 16′′. 21. Given A B 632, B ̊C 494, and ▲ A 20° 16′, to find the other parts, C being acute?

Answer, 4 C 26° 18′ 19′′, ▲ B 133° 25′ 41′′, and A C 1035.86. 22. Given A B 53′9, A C 46° 21′, and ▲ B 58·16, to find the other parts ? Answer, ZA 38° 58′, ▲ C 82° 46′, and B C 34·16. 23. Given A B 2163, B C 1672, and ▲ C 112° 18′ 22′′, to find the other parts?

Answer, A C 877.2, 4 B 22° 2′ 16′′, and 2 A 45° 39′ 22′′. 24. Given A B 496, B C 496, and B 38° 16', to find the other Answer, A C 325'1, 4A 70° 52', and 4 C 70° 52'.

parts?

25. Given A B 428, 4 C 49° 16', and A C + BC 918, to find the other parts, B being obtuse?

Answer 4 A 33° 44′ 48′′, ▲ B 91° 59′ 12′′, A C 564 49, and BC 353.5.

26. Given A C 126, 4 B 29° 46', and A B other parts?

B C 43, to find the

Answer, A 55° 51′32′′, ▲ C 94° 22′ 28′′, À B 253.54, and B C 210.54.

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

Answer, B 83° 23′ 47′′, 4 C 43° 19′ 53′′, and B C 1482-16. 28. Given A B 821.9, A C 640′3, and ▲ A 80° 24′, to find the other parts?

Answer, B 41° 26' 18", 4 C 58° 9′ 42′′, and B C 953.915. 29. Given A B 67·4, B C 53'11, and ▲ B 93° 26′ 44′′, to find the other parts?

Answer, 4 A 36° 54′ 23′′, ▲ C 49° 38′ 53′′, and A C 88·282, 30. Given A C 29674, B C 31283, and ▲ C 121° 5′ 38′′, to find the other parts?

Answer, A 30° 18′ 25′′, ▲ B 28° 35′ 57′′, and A B 53084'5. 31. Given AB 73, A C 100, and 4 A 2° 14′ 31′′, to find the other parts?

Answer, B 171° 43′ 59′′, C 6° 1′ 30′′, and B C 27.2062.

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 CBA 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.

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 A B =