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: rad

To compute 2 C. Here B C is the middle part, and A B and 2 C are adjoining extremes; hence the equation is rad. sin B C = tan A B.cot C, and the proportion to find 2 C is tan A B : rad : : sin B C : cot C. Or tan A B 29° 41' 39" 90756034

10.000000
: : sin B C 116 30 43

9.951746
; cot C
32 30 22

10:195712 EXAMPLES FOR EXERCISE. 1. In the right angled spherical triangle ABC, given AB 118° 21' 4", and 2 A 23° 40 12", to find the other parts ?

Answer, A C 116° 17' 55", Z C 100° 59' 26", and B C 21° 5' 42".

2. In the right angled spherical triangle A B C, given A B 53° 14' 20', and 4 A 91° 25' 53", to find the other parts ?

Answer, AC 91° 4' 9', 2 C 53° 15' 8", and B C 91° 47' 11". 3. In the right angled spherical triangle A B C, given A B 102° 50' 25'', and 4 A 113° 14' 37", to find the other parts ?

Answer, A C 84° 51' 36", 2 C 101° 46' 57", and B C 113° 46' 27".

4. In the right angled spherical triangle A B C, given A B 48° 24' 16'', and B C 59° 38' 27", to find the other parts ?

Answer, A C 70° 23' 49'', A 66° 20' 40", and 2 C 52° 32' 55".

5. In the right angled spherical triangle ABC, given A B 151° 23' 9', and B C 16° 35' 14'', to find the other parts ?

Answer; A C 147° 16' 51", 2C 117° 37' 21", and 2 A 31° 52' 50%.

6. In the right angled spherical triangle A B C, given A B 773° 4' 31", and A C 86° 12' 15", to find the other parts ?

Answer, B C 76° 51' 20", LA 77° 24' 23", and 2 C 73° 29' 40".

7. In the right angled spherical triangle A B C, given AC 118° 32' 12", and A B 47° 26' 35", to find the other parts?

Answer, BC 134° 56' 20", A 126° 19' 2", and 4 C 56° 58' 44".

8. In the right angled spherical triangle A B C, given Ą C 91° 50' 23", and A B 92° 17' 26', to find the other parts ?

Answer, BC 36° 33' 29", 2 A 36° 34' 50", and C 91° 22'00". 9. In the right angled spherical' triangle A B C, given A B 138° 25' 34", and A C 49° 27' 16', to find the other parts ?

Answer, B C 150° 20' 8", 2 A 139° 21' 36', and 2 C 119° 9' 34".

10. In the right angled spherical triangle A B C, given AC 68° 14' 20", and 2 C 70° 21' 15', to find the other parts ?

Answer, B C 40° 6' 19", A B 61° 0' 22", and 2 A 43° 55' 2". 11. In the right angled spherical triangle A B C, given AC 118° 25' 21", and 2 C 53° 27' 46", to find the other parts ?

Answer, B C 132° 16' 22'', A B 44° 57' 38", and 2 A 44° 57' 38". 12. In the right angled spherical triangle A B C, given A C 53° 25' 31", and 4 A 124° 26' 7'', to find the other parts ?

13. In the right angled spherical triangle A B C, given AC 102° 15' 27", and B C 49° 13' 18", to find the other parts ? Answer, A B 108° 58' 9", 4 A 50° 47' 47", and 2 C 101° 35' 21".

14. In the right angled spherical triangle ABC, given 2C 38° 14' 3", and 4 A 59° 20°"", to find the other parts ?

Answer, B C 34° 30' 11", A B 24° 3' 2", and A C 41° 11' 17". 15. In the right angled spherical triangle A B C, given 2 C 171° 4' and 4 A 92° 6', to find the other parts ? Answer, A C 76° 30' 37", A B 171° 18' 56'', and B C 103° 38' 5711.

16. In the right angled spherical triangle ABC, given 2C 90° 18' 18'', and 4 A 93° 17' 20', to find the other parts ?

Answer, A C 89° 58' 56", A B 90° 18' 20", and B C 93° 17' 20''. 17. In the right angled spherical triangle A B C, given A B 40° 18' 23'', and A C 100° 3'7", to find the other parts ?

Answer, 4 A 98° 38' 53'', Z C 41° 4' 6", and B C 103° 13' 59". 18. In the right angled spherical triangle A B C, given AC 61° 3' 22'', and 4 A 49° 28' 12", to find the other parts ?

Answer, A B 49° 36' 6", 2 C 60° 29' 19", and B C 41° 41' 32". 19. In the right angled spherical triangle A B C, given A B 29° 12' 50', and 2 C 37° 26' 21", to find the other parts ?

Answer, ambiguous, 4 A 65° 27' 58" or its supplement, AC 53° 24' 13" or its supplement, B C 46° 55' 9" or its supplement.

20. In the right angled spherical triangle A B C, given AB 54° 21' 35", and 2 C 61° 2' 15", to find the other parts ?

Answer, ambiguous, BC 129° 28' 28" or its supplement, AC 111° 44' 34" or its supplement, and 2 A 123° 47' 44'' or its supplement.

21. In the right angled spherical triangle A B C, given A B 100° 10' 3", and C 90° 14' 20", to find the other parts ?

Answer, ambiguous, AC 100° 9' 55" or its supplement, BC 1° 19' 53' or its supplement, and 2 A 1° 21' 8"' or its supplement.

22. In the right angled spherical triangle A B C, given A B 121° 26' 25'', and Z C 111° 14' 37", to find the other parts ?

Answer, ambiguous, 2 A 136° 0' 3'' or its supplement, A C 66° 15' 38" or its supplement, and B C 140° 30' 56' or its supplement.

APPLICATION OF THE FORMULÆ FOR RIGHT ANGLED

SPHERICAL TRIANGLES TO THE SOLUTION OF CASES RELATIVE TO QUADRANTAL TRIANGLES.

A quadrantal triangle is a spherical triangle one of whose sides is a quadrant.

Let ABC or A B' C be a quadrantal triangle, AC being the quadrantal side, on C B' or CB produced, let C D be taken

B

B'

equal to a quadrant, and let A D be an arc of a great circle passing through A and D. Then the angles CAD, CDA (or B D A) and B'DA will be right angles, and A D will be the measure of the angle C. The angles D A B, D A B' will respectively be complements of C AB, CA B, and CB, C B' will also be respectively complements of B D, B' D. Hence the different parts of the quadrantal triangles B' AC, BAC may be determined from the corresponding parts of the right angled triangles A D B, A D B'.

EXAMPLE I. In the triangle A B’ C, given A C 90°, the angle C A B' 112° 2' 9", and A B 67° 3' 14", to find the other parts ?

Let CD be a quadrant, then as CAD will be a right angle, D A B' will be 22° 2' 9". Hence D B' will be acute; and as A B' is acute, the angle B' and A D, the measure of the angle C, will also be acute.

To compute A D, or the measure of 2 C. Equation, rad . cos D A B' = cot A B' . tan A D. Or cot A B 67° 3' 14"

9.626715

10.000000
: : cos D A B’ 22 2 9

9:967057
: tan AD
65 27 9

10:340342
To compute D B', the complement of DC.
Equation, rad . sin D B' = sin D A B'.sin A B'.
Or rad

10'000000
: sin D A B 22° 2' 9/1

9:574246 :: sin A B' 67 3 14

9.964199 : sin D B' 20 12 44

9.538445 90

.

: rad

B'C 110 12 44

To find the angle B'. Equation, rad . cos A B' = cot DA B' . cot B'. Or cot D A B' 22° 2' 9.....

10392809

10:000000
:: cos A B'
67 3 14

9590915

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: rad

: cot 2 B'

81 1 58

9.198106

EXAMPLE II. In the triangle A B C, given A C 90°, A B' 79° 18' 40", and C B 123° 16' 3', to find the other parts ?

Let C D be a quadrant, then D B', the complement of CD, is To find the angle B'. Equation, rad . cos 2 B' = cot A B'tan D B'. Or rad

10.000000 : cot A B 79° 18' 40

9:275889
:: tan D B 33 16 3

9.816946
: cos LB
82 53 12

9.092835

To find A D, or the measure of the angle C.
Equation, rad . cos A B' = cos D B'. cos DA.
Or cos D B'33° 16' 3''.....

9.922268
: rad ....

10.000000
:: cos A B 79 18 40

9.268288
: cos A D
77 11 0

9.346020

: rad

To find the angle D A B'. Equation, rad . sin D B' = sin A B', sin D A B'. Or sin A B' 79° 18' 40"

9.992398

10.000000 ::sin D B 33 16 3

9739215
: sin D A B’ 33 56

90746817
90
CA B' 123 56 1

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1. In the quadrantal triangle A B C (see the last figure) AC being the quadrantal side, given A B 67° 3' and 2 A 49° 18', to find the other parts ?

Answer, 6 C 60° 48' 54", B C 53° 5' 46'', and B 108° 32' 27".

2. Given 2 A 118° 40' 36", and B C 113° 2' 28", to find the other parts?

Answer, A B 54° 38' 57", LC 51° 2' 35", and 2 B 72° 26' 21". 3. Given C 69° 13' 46", A 72° 12' 4", to find the other parts?

Answer, A B 70° 8' 39", B C 73° 17' 29", and 2 B 96° 13' 23". 4. Given B C 86° 14' 40', and 2 A 37° 12' 20", to find the other parts?

Answer, AB 4° 43'2", 2 B 142° 42' 2", and 2 C 2° 51' 23". 5. Given C 60° 41' 30", and B C 78° 12' 19", to find the other parts?

Answer, A B 61° 22' 7", LA 76° 31' 59", and 2 B 96° 32' 45'. 6. Given B C 118° 32' 16'', and A B 67° 48' 40', to find the other parts ?

Answer, 4 C 64° 32' 21", 6 A 121° 3' 40", and 4 B 77° 11' 6". 7. Given B C 59° 3' 42", and A B 61° 4' 19'', to find the other parts?

Answer, 2 C 55° 15' 0", Z B 110° 9' 10", and 4 A 52° 48'-46". 8. Given 4 B 104° 41' 17", and B C 73° 21' 6'', to find the other parts ?

Answer, 4 A 67° 56' 13', 2 C 47° 32' 39'', and A B 49° 42' 18''.

9. Given 2 A 21° 39' 48'', and < C 53° 26' 45', to find the other parts ?

Answer, 2 B 123° 36' 32", B C 26° 18' 40", and A B 74° 41' 35".

10. Given 2 B 94° 29' 54', and B C 56° 31' 26", to find the other parts?

Answer, 4 A 56° 13' 28", 2 C 81° 53' 0", and A B 83° 14' 11".

11. Given B C 18° 28' 1", and 4 C 93° 18' 32", to find the other parts ?

Answer, LA 78° 26' 54", 2 B 89° 20' 16'', and A B 93° 14' 30".

12. Given A B 96° 32' 18", and B C 85° 32'4", to find the other parts?

Answer, 2 A 35° 30' 19", 2C 96° 33' 29", and 4 B 89° 29' 14".

APPLICATION OF TRIGONOMETRICAL FORMULÆ TO THE NUMERICAL COMPUTATION OF THE DIFFERENT PARTS OF OBLIQUE ANGLED SPHERICAL TRIANGLES.

All the cases of oblique angled spherical triangles may be solved by the formulæ for right angled ones, except when the three sides are given to find the angles, or the three angles given to find the sides. For the solution of these two cases, we have the following rules.

To find any angle of a spherical triangle when the three sides are given.

Rule 1. From half the sum of the three sides subtract the side opposite to the required angle, then add together the log cosecants of the other two sides, (rejecting 10 from each of their indexes) and the log sines of the half sum, and remainder; half the sum of these four logarithms will be the log cosine of half the required angle.

RULE 2. From half the sum of the three sides, subtract each of the sides containing the required angle, then add together the log cosecants of these two sides, (rejecting 10 from each of their indexes) and the log sines of the two remainders ; half the sum of these four logarithms will be the log sine of half the required angle.

Remark. When the required angle is large, the first of these rules

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