CASE V. The Angles B and C, and the Leg. A C, given; Fig. XII. To find the Hypotenuse B C. Cafe V. PROPORTION. Ass B: Leg. A C:: Radius: Hypotenuse BC. [By Axi. II. CASE VI. The Hypotenuse BC, and the Leg. A C, given, Fig. XII. To find the other Leg. A B. PROPORTIONS. 1.) As Leg. BC: Rad.:: Leg. AC:s B. [By Cafe I. 2.) As 1 B: Leg. AC:: Rad. : Leg. A B. [By Cafe L. Cafe VI. i L : Fig. XII. CASE VII. The two Legs, B A and C A, given; To find the PROPORTION S. 1.) As Leg. A B: Rad.:: Leg. AC:t B. [By Cafe I. The End of the Seven Cafes of Right-angled Plain Triangles. CHAP. VI. Of the Solution (or Mensuration) of Oblique-angled Fig. XII. T : ASE I. Two Sides, DC, and CB, with the Angle at D, Fig. XII. opposite to C B, given; To find the Angle at B, oppofite to CD. Cafe I. PROPORTION. is Log. CB: SD:: Log. CD, :s B. [By Axiom II. But here it must be known whether the Angle fought be Acute or Obtufe. ASE II. Two Sides, DC, and BC, with the Angle C, com- Fig. XII? prehended by them, given; To find the other Angles B or D. The given Angle C being above 90 Deg. viz. 115.18 Deg. abstract it from 180 Deg. the Remainder is 64.82 Deg. ✓hich must be made use of in the Calculation: And then, This the PROPORTION. is half the Sum of the given Sides DC and BC, Is to half the Difference of those Sides; o is the Tangent of half the Sum of the Angles at B and D,. To the Tangent of half the Difference of those Angles. By Axiom III. Then, ( To the half Sum of these Angles D and B, if you Add the half Difference thus found, you will have the Greater Angle; but eing substracted, it will give you the Leffer Angle. Cafe II.. OPE Fig. XII. CASE VII. The two Legs, B A and C A, given; To find th Cafe VII. Hypotenuse B C. PROPORTION S. 1.) As Leg. A B: Rad. :: Leg. AC:t B. [By Cafe I. Is to Leg. A C, 142.72 Parts 12.15448 So is Rad. Sine 90 Deg. To the Hypotenuse B C, 274.93 Parts The End of the Seven Cafes of Right-angled Plain Triangles. 10. 243929 CHAP. VI. Of the Solution (or Mensuration) of Oblique-angled HE Triangle which I shall make use of in the Solution XII. TH Diagram the Five CASES of Oblique-angled Triangles, shall S. that noted with S; whose Sides and Angles are CASE I. Two Sides, DC, and CB, with the Angle at D, Fig. XII. opposite to C B, given; To find the Angle at B, oppofite to CD. Cafe I. PROPORTION. As Log. CB:SD :: Log. CD, :s B. [By Axiom II. But here it must be known whether the Angle fought be Acute or Obtufe. CASE II. Two Sides, DC, and BC, with the Angle C, com- Fig. XII? prehended by them, given; To find the other Angles B or D. The given Angle C being above 90 Deg. viz. 115.18 Deg. Substract it from 180 Deg. the Remainder is 64.82 Deg. Which must be made use of in the Calculation: And then, This is the : PROPORTION. As half the Sum of the given Sides DC and BC, Is to half the Difference of those Sides; So is the Tangent of half the Sum of the Angles at B and D,. To the Tangent of half the Difference of those Angles. By Axiom III. Then, To the half Sum of these Angles D and B, if you Add the half Difference thus found, you will have the Greater Angle; but being substracted, it will give you the Leffer Angle. Cafe.II.. こ OPE |