TRIGONOMETRY. 



fubtrafted from the fum of the whole fine, and the cofine of 

 B, the remainder, is the fine of C. The former example, 

 therefore, is eafily accommodated to the prefent cafe. 



8. Given the obhque angles B, 77° 45', and C, 23" 30' ; 

 to iind the leg adjacent to the other, A C. From problem 

 the fixth, it is evident that the fine of C is to be fubtrafted 

 from the fum of the whole fine, and the cofine of B ; and 

 that the remainder is the cofine of A C. The example of 

 the fixth problem may be eafily applied to this. 



9. Given tlie leg A C, 57° 49', and the adjacent angle C, 

 23° 30' ; to find the oppofite leg A B. Since A C is a 

 mean part, and C and A B conjunft parts, the produft of 

 the whole fine, by tlie fine of A C, is equal to the produft 

 of the cotangent of C, and the tangent of A B. 



Therefore from the whole fine 10.0000000 

 Sine of A C 9.9275490 



Sum 

 Subtraft cotangent of C 



19.9275490 

 10.3616981 



Remains tangent of A B 9.5658519 Towhich 

 the neareft correfponding number, in the tables, is 20° 12'. 



10. Given the leg A B, 20' 12', and the oppofite angle C, 

 23"^ 30'; to find the adjacent leg A C. From the fum of the 

 cotangent of C, and the tangent of A B, fubtraft the whole 

 fine ; the remainder is the fine of A C. 



1 1 . Given the legs A B, 20° 1 2', and A C, 57° 49' ; to find 

 the angle C, oppofite to one of them. From the fum of 

 the whole fine, and fine of A C, fubtraft the tangent of B A ; 

 the remainder is the cotangent of C. 



12. Given the hypothenufe B C, 60°, and the obhque 

 angle C, 23° 30'; to find the adjacent leg A C. Since C is 

 a middle part, and B C and A C conjoint parts ; the pro- 

 duft of the whole fine into the cofine of C, will be equal to 

 the produft of the cotangents of A C and B C. 



Therefore from the whole fine 10.0000000 

 Cofine of C 9.9623978 



Sum 

 Subtraft cotangent of B C 



19.9623978 

 9.7614394 



Remains tangent of A C 10.2009584 The near- 

 eft number correfponding to which, in the tables, is 57° 49'. 



13. Given the leg A C, 57^49', and the adjacent angle C, 

 23° 30' ; to find the hypothenufe B C. 



From the fum of the whole fine, and the cofine of C, fub- 

 traft the tangent of A C ; the remainder is the cotangent 

 of BC. 



14. Given the hypothenufe B C, 608, and the leg A C, 

 57° 49'; to find the adjacent angle C. 



From the fum of the cotangent of B C, and tangent of 

 A C, fubtraft the whole fine ; and the remainder is the co- 

 fine of C. 



15. Given the hypothenufe B C, 60", and one angle C, 

 23° 30' ; to find the other, B. 



Since B C is the middle part, and B and C disjunft parts, 

 the produft of the whole fine, into the cofine of B C, will 

 be equal to the produft of the cotangents of B and C ; and 



Therefore from the whole fine 10.0000000 

 Cofine of B C 9.6989700 



Sum 

 Subtraft cotangent of C 



19.6989700 

 10.3616981 



Remains cotangent of B 9-3372719 The neareft 

 correfponding number to which, in the tables, is 12° 16'; 

 therefore B = 77° 44'. 



16. Given the oblique angles B, 77° 44', and C, 23° 30'- 

 to find the hypothenufe B C. From the fum of the co'- 

 tangents of C and B, fubtraft the whole fine ; the remainder 

 is the cofine of B C. 



Solution of oblique-angled fpherical Triangles. ^-i. In an ob- 

 lique-angled fpherical triangle ABC, {Jig. 5.) two fides, 

 A B and B C, being given, together with an angle, A, 

 oppofite to one of them, to find the other, C : the rule is 



As the fine of the fide B C is to the fine of the oppofite 

 angle A ; fo is the fine of the fide B A to the fine of the 

 oppofite angle C. 



Suppofe, for example, B C, 39° 29' ; A, 43° 20' ; B A, 

 66° 45' ; then will 



Sine of B C 9.8033572 

 Sine of A 9.8364771 

 Sine of BA 9.9632168 



■ 19-7996939 

 Sine of C 9-99<533<57 The neareft correfpond- 

 ing number to which, in the tables, is 82° 34'. 



2. Given two angles C, 82° 34', and A, 43° 20', together 

 with the fide A B, 66° 45', oppofite to one of them, C ; 

 to find the fide B C oppofite to the other of them, A ; fay, 

 as the fine of the angle C is to the fine of the oppofite fide 

 A B ; fo is the fine of the angle A to the fine of the oppo- 

 fite fide B C. The former example may fuffice for the 

 prefent cafe. 



3. Given two fides A B, 66° 45', and B C, 39° 29', to- 

 gether with an angle oppofite to one of them A, 43° 20' ; to 

 find the angle included by them, B. Suppofe the angle C to 

 be acute, fince the other. A, is alfo acute, the perpendicular 

 B E falls within the triangle. In the right-angled triangle 

 ABE, therefore, from the given angle A, and fide A B, 

 find the angle ABE. Since B E is affumed as a lateral 

 part in the triangle A E B, the angle E B C is a middle 

 part, and the fide B C muft be a conjoint part : the cofine 

 of the angle E B C will be found by fubtrafting the co- 

 tangent of A B from the fum of the cofine of the angle 

 ABE, and the cotangent of B C. If then the angles 

 ABE and E B C be added together, or in cafe the per- 

 pendicular fall without the triangle, be fubtrafted from each 

 other, you will have the angle required B. 



E. gr. Whole fine 10.0000000 



Cofine of A B 9.5963154 



Sum 

 Cotangent of A 



19.5963154 

 10.0252805 



Cotangent of ABE 9.5710349 The neareft 



number correfponding to which, in the tables, is 20° 26'. 

 ABE, therefore, is 69° 34'. 



Cofine of A B E 9.5429713 



Cotangent of B C 10.0841529 



Sum 



Cotangent of A B 



19.6271242 

 9.6330985 



Cofine of E B C - 9.9940257 The neareft 



number correfponding to which, in the tables, is 9° 29' ; 

 therefore A B C = 79° 3'. 



4. Given two angles A, 43° 20', and B, 79° 3', toge- 

 ther with the adjacent fide A B, 66° 45' ; to find the fide 

 B C oppofite to one of them. 



From one of the given angles B, let fall a perpendicular 



E B to the unknown fide of A C ; and in the right-angled 



triangle ABE, from the given angle A, and hypothenufe 



A B, find the angle ABE; which, fubtrafted from the 



M m 2 angle 



