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

By the First Rule.
AB 59° 16' 23"
в с 70

4 18 cosec *026817
A C 63 21 27 cosec •048749
1)192 42 8
96 21
4 sin

9.997326
37 4 41 sin 9.780247

1)19.853139
32 23 17 COS 9.926569

2
64 46 34 LC

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

LA 38° 19' 18'' supplement 141°40 42"
LB 48 0 10 supplement 131 59 50
LC 121 8 6

6 supplement 58 51 54 141° 40 42"

131° 59' 50% 131 59 50 cosec •128908 141 40 42 cosec .207555 58 51 54 cosec 067551

58

51 54 cosec •067551 2)332 32 26

2)332 32 26 166 16 13 sin 9:375375

166 13. sin 9:375375. 24 35 31 sin 9:619253

34 16 23 sin 9.750614 2)19:191087

2)19.401095 66 47 37] cos 9:595543 59 52 47] cos 9:700547 2

2 133 35 15 sup ZA

119 45 35 sup 2 B 46 24 45 L. A

60 14 25 LB

16

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90 58 46 sup LC

89 1 14 LC 3. In the triangle A B C, given A B 91° 3' 26", AC 40° 36' 37", and ZA 58° 31', to find the other parts ?

Let C D (see the first of the adjoining figures) be a great circle drawn through C, perpendicular to A B. Then in the right angled tri

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C angle ACD are given A C and 2 A to find the other parts. Now as 2 A are both given acute, AD, DC, and A

D

B
LAC D are all acute.

D
To find A D.
Radius . cos ZA= tan AD.cot A C.

Or cot A C 40° 36' 37" 10-066810
: rad

10.000000
:: cos 4 A 58 31 0

9.17879
i tan AD 24 y 19

9.651069
AB 3 26
DB 66 56

To find / ACD.
Radius . cos AC = cot · A. cot A C D.
Or cot Z A 58°31' 0'.... 9:787036

10.000000
: : cos A C

40 36 37. 9.880330 : cot 2 CD 38 53 34... 10.093294 As D C and D B are both acute, BC and the 2 BCD, C B D, are also acute.

To find BC.
As cos A D 24° 197 9.960317
: cos D B
66 56

9:593032
:: cos AC
40 36 37.... 9.880330

19:473362

91

: rad

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To find Z BCD.
As tan A D 24° al 19/

9.651066
: tan B D 66 56 7

10:370786 :: tan ACD 38 53 34

9:906706

20-277492
: tan BCD 76 42 6

...... 10:626426
115 35 40 ACB, or sum of ACD and BCD.

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

Answer, B C 23° 57' 13', 2 B 91° 26' 44", and 'C 102° 5' 54''. 2. Given 2 A 81° 38' 17", 2 B 70° 9' 38', and 2 C 64° 46' 32", to find the sides?

Answer, A B 59° 16' 23", B C 70° 4' 18', and 2 C 63° 21' 271. 3. Given A B 81° 12', A C 84° 16', and 4 C 80° 28', to find the other parts ?

Answer, this example produces an ambiguous result. If the angle B be considered as acute, then 2 A = 97° 13' 45'', 2 B 83° 11' 24'', and B C 96° 13' 33'' ; but if B be considered as obtuse, then 4 A= 21° 16' 44", Z B 96° 48' 36'', and BC 21° 19' 29".

4. Given A B 64° 26', LA 49°, and Z B 52°, to find the other parts ?

Answer, A C 45° 56' 46", BC 43° 29' 49'', and 2 C 98° 28' 5'. 5. Given A B 96° 14' 50%,B C 93° 27' 34", and AC 100° 4' 26", to find the angles ?

Answer, 4 A 94° 39' 4", 2 B 100° 39' 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, L A 97° 35' 32', L B 85° 8'0', and 2 C 88° 34' 20'. 7. Given A B 78° 29' 35", BC76° 1' 43', and 4 B 82° 59' 26'', to find the other parts ?

Answer, AC 68° 14' 30', ZA 79° 23' 42", and 2 C 70° 10 24". 8. Given A B 67° 14' 28", B C 40° 18' 29'', and 2 A 34° 22' 17", to find the other parts?

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