When the three angles are given to find the sides.

RULE. Take the supplements of the measures of the given angles, as the sides of another triangle, and find the angles of that triangle by either of the preceding rules, and the supplement of the measures of these angles will be the sides of the proposed triangle.

When a side and two of the angles, or an angle and two of the sides, are given to find the other parts.

RULE. Let a perpendicular be drawn from an extremity of a given side, and opposite a given angle, or its supplement; there will then be formed two right angled triangles, the parts of which may be computed by Napier's Rules.

The following proportions however which were deduced at Prop. 16 and 17, Elements of Spherics, will often be found useful.

1. The sines of the sides of spherical triangles are proportional to the sines of their opposite angles.

2. The sines of the segments of the base, made by a perpendicular from the opposite angle, are proportional to the cotangents of their adjacent angles.

3. The cosines of the segments of the base are proportional to the cosines of the adjacent sides of the triangle.

4. The tangents of the segments of the base are proportional to the tangents of the opposite segments of the vertical angles.

5. The cosines of the angles at the base are proportional to the sines of the corresponding segments of the vertical angles.

6. The cosines of the segments of the vertical angles are proportional to the cotangents of the adjoining sides of the triangle.

EXAMPLES.

1. In the triangle ABC, given A B 59° 16′ 23′′, B C 70° 4' 18", and A C 63° 21′ 27′′, to find the other parts?

2. In the triangle A B C, given ▲ A 38° 19′ 18′′, 4 B 48° 0′ 10′′, C 121° 8' 6", to find the other parts?

3. In the triangle A B C, given A B 91° 3′ 26′′, AC 40° 36′ 37"",

and 4 A 58° 31′, to find the other parts ?

Let CD (see the first of the adjoining figures) be a great circle drawn through C, perpendicular to A B. Then in the right angled triangle A CD are given A C and A to find the other parts. Now as 4 A are both given acute, AD, DC, and ZAC D are all acute.

To find A D.

Radius. cos A tan AD. cot A C.

D

B

D

As D C and D B are both acute, BC and the BC D, C B D, are

1. In any triangle A B C, given A C 118° 2′ 14′′, 4 A 27° 22′ 34", and A B 120° 18′ 33′′, to find the other parts?

Answer, B C 23° 57′ 13′′, ▲ B 91° 26′ 44′′, and C 102° 5′ 54′′. 2. Given A 81° 38′ 17′′, 4 B70° 9′ 38′′, and ▲ C 64° 46′ 32", to find the sides?

Answer, A B 59° 16′ 23′′, B C 70° 4′ 18′′, and 4 C 63° 21′ 27′′. 3. Given A B 81° 12′, A C 84° 16′, and ▲ C 80° 28′, to find the other parts?

Answer, this example produces an ambiguous result. If the angle B be considered as acute, then ▲ A = 97° 13′ 45′′, 4 B 83° 11′ 24′′, and B C 96° 13′ 33′′; but if B be considered as obtuse, then 4 A = 21° 16' 44", 4 B 96° 48′ 36", and B C 21° 19' 29".

4. Given A B 64° 26′, ▲ A 49°, and ▲ B 52°, to find the other parts?

Answer, A C 45° 56′ 46", BC 43° 29′ 49", and 4 C 98° 28′ 5′′. 5. Given A B 96° 14′ 50′, B C 93° 27′ 34′′, and A C 100° 4′ 26′′, to find the angles?

Answer, 4 A 94° 39′ 4", 4 B 100° 32′ 19′′, and 4 C 96° 58′ 36′′. 6. Given A B 89° 12′ 20′′, B C 97° 30′ 0′′, and AC 85° 16′ 48′′, to find the angles?

Answer, 4 A 97° 35′ 32′′, 4 B 85° 8' 0", and 4 C 88° 34′ 20′′. 7. Given A B 78° 29′ 35′′, BC 76° 1' 43", and 4 B 82° 59′ 26′′, to find the other parts?

Answer, A C 68° 14′ 30′′, ▲ A 79° 23′ 42′′, and 4 C 70° 10′ 24′′. 8. Given A B 67° 14′ 28′′, B C 40° 18′ 29′′, and ▲ A 34° 22′ 17′′, to find the other parts?