: 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 9.951746 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 9:967057 10:340342 10'000000 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 9590915 : 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 9.816946 9.092835 To find A D, or the measure of the angle C. 9.922268 10.000000 9.268288 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 90746817 1.. 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 |
