The remaining angles may now be found by Problem 4.

PRACTICAL PROBLEMS.

Let ABC represent any oblique-angled triangle. 1. Given, AB 697, the angle A 81° 30′ 10′′, and the angle B 40° 30' 44", to find the other parts.

Ans. AC, 534; BC, 813; and LC, 57° 59′ 6′′.

=

2. If AC720.8, LA 70° 5′ 22′′, LB = 59° 35′ 36", required the other parts.

Ans. AB, 643.2; BC, 785.8; and LC, 50° 19' 2". 3. Given, BC 980.1, the angle A 7° 6′ 26′′, and the angle B 106° 2′ 23′′, to find the other parts.

Ans. AB, 7283.8; AC, 7613.1; and LC, 66° 51' 11".

4. Given, AB 896.2, BC 328.4, and the angle €113° 45' 20", to find the other parts.

Ans. {

AC, 712; LA, 19° 35′ 46′′; and LB, 46° 38′ 54′′.

5. Given, AC = 4627, BC= 5169, and the angle A

70° 25' 12", to find the other parts.

Ans.

=

{ AB, 4828; LB, 57° 29′ 56′′;

and C, 52° 4′52′′.

6. Given, AB 793.8, BC 481.6, and AC 500.0, to find

the angles.

Ans.

LA, 35° 15′ 32"; LB, 36° 49' 18"; and LC, 107° 55' 10".

7. Given, AB 100.3, BC 100.3, and AC 100.3, to find

the angles.

Ans.

{

The angle A, 60°; the angle B, 60°; and the angle C, 60°.

8. Given, AB 92.6, BC 46.3, and AC 71.2, to find the angles.

Ans. (LA, 29° 17′ 22′′; LB, 48° 47′ 30′′; and LC,

9. Given, AB 4963, BC 5124, and AC 5621, to find the angles.

Ans. {LA, 570 30′ 28′′; LB, 67° 42′ 36′′; and \_C,

54° 46' 55".

10. Given, AB 728.1, BC 614.7, and AC 583.8, to find the angles.

Ans.

La

{LA=54032′52′′, LB = 50° 40′ 58′′, and __C

74° 46' 10".

11. Given, AB 96.74, BC 83.29, and AC 111.42, to find the angles.

Ans.

LA=46° 30′ 45′′, LB=76° 3′ 46′′, and LC = 57° 25' 29".

{L

12. Given, AB 363.4, BC 148.4, and the angle B 102° 18' 27", to find the other parts.

A 20° 9′ 17′′, the side AC-420.8, and C

LA

=

=

57° 32′ 16′′.

13. Given, AB 632, BC 494, and the angle A 20° 16', to find the other parts, the angle being acute.

=

Ans. {LO 26° 18′ 19′′, _ B = 133° 25′ 41′′, and

AC1035.7.

14. Given, AB 53.9, AC 46.21, and the angle B 58° 16', to find the other parts.

Ans. LA 38° 58', LC= 82° 46', and

=

BC=34.16.

15. Given, AB 2163, BC 1672, and the angle C 112° 18' 22", to find the other parts.

Ans. AC, 877.2; LB, 22° 2′ 16′′; and LA, 45° 39′ 22′′. 16. Given, AB 496, BC 496, and the angle B 38° 16', to find the other parts.

Ans. AC, 325.1; LA, 70° 52′; and LC, 70° 52'. 17. Given, AB 428, the angle C 49° 16', and (AC+ BC) 918, to find the other parts, the angle B being

obtuse.

Ans.

{

The angle A= 38° 44′ 48′′, the angle B = 91° 59′ 12′′, AC=564.5, and BC=353.5.

18. Given, AC 126, the angle B 29° 46′, and (AB— BC) 43, to find the other parts.

Ans.

{

The angle A=55° 51′ 32′′, the angle C=94° 22' 28", AB = 253.05, and BC= 210.05.

19. Given, AB 1269, AC 1837, and the angle A 53° 16' 20", to find the other parts.

Ans.

B=83° 23′ 47′′, C= 43° 19' 53", and BC

SECTION III.

APPLICATION OF TRIGONOMETRY TO MEASURING HEIGHTS AND DISTANCES.

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 employed in the determination of angles where anything like precision is required.

The following problems 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.

PROBLEM I.

Being desirous of finding the distance between two distant objects, C and D, I measured a base, AB, of 384 yards, on the same horizontal plane with the objects C

and D. At A, I found the angles DAB = 48° 12′, and CAB 89° 18'; at B, the angles ABC 46° 14′, and ABD 87° 4′. It is required, from these data, to compute the distance between C and D.

From the angle CAB, take the angle DAB; the remainder, 41° 6, is the angle CAD. To the angle DBA, add the angle DAB, and 44° 44′, the supplement of the sum, is the angle ADB. In the same way the angle ACB, which is the supplement of the sum of CAB and CBA, is found to be 44° 28'. Hence, in the triangles ABC and ABD, we have

Then, in the triangle CAD, we have given the sides CA and AD, and the included angle CAD, to find CD; to compute which we proceed thus:

The supplement of the angle CAD, is the sum of the angles ACD and ADC;