be: Making the Leg BC Radius, the Proportions will To find the Leg AB. As Secant AC, 54° 30′ AC Hyp. AC, 25 :: Tangent ACB, 54° 30' To find the Leg BC. As Secant ACB, 54° 30′ : Hyp. AC, 25 :: Radius The Logarithms of the four last Proportions being looked out, and added and subtracted according to the Rule, the result will be found to be the same as the twe first Proportions. By Natural Sines. This CASE may be solved by Natural Sines,* according to the following Proportions: As Unity or 1; Is to the length of the hypothenuse; So is the Natural Sine of the smallest Angle; To the length of the shortest Leg. Or, So is the Natural Sine of the largest Angle; To the length of the longest Leg. Or, which is the same thing, Multiply the Natural Sines of the two Angles by the Hypothenuse, the Products will be the length of the two Legs. first Term, the Remainder will be the Log. of the fourth Term, which seek in the Tables and find its corresponding Number or Degrees and Minutes. See the Introduction to the Table of Logarithms; which should be attentively studied by the Learner before he proceeds any further. Note. The Logarithm for Radius is always 10, which is the Logarithmic Sine of 90°, and the Logarithmic Tangent of 45°. The preceding PROPOSITIONS and RULES being duly attended to, the solution of the following CASES of Rectangular Trigonometry will be easy. CASE I. The Angles and Hypothenuse given to find the Legs. Fig. 39. In the Triangle ABC, given the Hypothenuse AC 25 Rods or Chains; the Angle at A 35° 30', and consequently the Angle at C 54° 30': to find the Legs. Making the Hypothenuse Radius, the Proportions will be; To find the Leg AB. As Radius Hyp. AC, 25 To find the Leg BC. 10.00000 : Hyp. AC, 25 10.00000 :: Sine ACB, 54° 30′ 9.91069 :: Sine CAB, 35° 30' 9.76395 Note. When the first Term is Radius, it may be Subtracted by cancelling the first figure of the Sum of the other two Terms. Making the Leg AB Radius, the Proportions will be: To find the Leg AB. As Secant CAB, 35° 30′ : Hyp. AC, 25 :: Radius ་ To find the Leg BC. As Secant CAB, 35° 30′ : Hyp. AC, 25 :: Tangent CAB, 35° 30* The Logarithms of the four last Proportions being looked out, and added and subtracted according to the Rule, the result will be found to be the same as the two first Proportions. By Natural Sines. This CASE may be solved by Natural Sines,* according to the following Proportions: As Unity or 1; Is to the length of the hypothenuse; So is the Natural Sine of the smallest Angle; To the length of the shortest Leg. Or, So is the Natural Sine of the largest Angle; To the length of the longest Leg. Or, which is the same thing, Multiply the Natural Sines of the two Angles by the Hypothenuse, the Products will be the length of the two Legs. Note. The third Decimal figure in the first Product being 7, the preceding figure may be called one more than it is, viz. 2. And, whenever in any Product, &c. there are more places of Decimals than you wish to work with, if the one at the Right Hand of the last which you wish to retain is more than 5, add a Unit to the last; because a greater number than 5 is more than half. As the Table of Artificial or Logarithmic Sines, Tangents and Secants contained in this Book, is calculated only for every 5 Minutes of a Degree, whenever any Question is to be solved where the Minutes cannot be found in that Table; or where the length of the Hypothenuse is such a number as cannot be found in the Table of Logarithms for Numbers, the Question may be solved by Natural Sines, as above taught. CASE II. The Angles and one Leg given, to find the Hypothenuse and the other Leg. Fig. 40. In the Triangle ABC, given the Leg AB 325, the Angle at A 33° 15', and the Angle at C 56° 45'; to find the Hypothenuse and the Leg BC. Note. Reject the first figure, which is the same as subtracting Radius, and seek the numbers corresponding to the other figures. Making the Leg BC Radius, the Proportions will be; To find the Hypothenuse. As Tang. ACB, 56° 45′ : Leg AB, 325 :: Sec. ACB, 56° 45′ To find the Leg BC. As Tang. ACB, 56° 45′ : Leg AB, 325 :: Radius Making the Hypothenuse Radius, the Proportions will be: To find the Hypothenuse. As Šine BCA, 56° 45′ : Leg AB, 325 :: Radius : Hyp. 388.6 To find the Leg BC. As Sine BCA, 56° 45′ : Leg AB, 325 :: Sine BAC, 33° 15′ Note. If the Leg BC had been given, instead of the Leg AB, the Proportions would have been the same mutatis mutandis. By Natural Sines. To solve this CASE by Natural Sines, institute the following Proportions: To find the Hypothenuse. As the Natural Sine of the Angle opposite the given Leg; Is to the length of the Leg; So is Unity or 1; To the length of the Hypoth enuse. Or, which is the same thing, Divide the given Leg by the Natural Sine of its opposite Angle, and the Quotient will be the Hypothenuse. To find the other Leg. As the Natural Sine of the Angle opposite the given Leg; Is to the length of the given Leg; So is the Natural Sine of the Angle opposite the other Leg; To the length of the other Leg. EXAMPLE. Given Leg 325. Nat. Sine of 56° 45', the Angle opposite the given Leg 0.83629 Nat. Sine of 33° 15′, the Angle opposite the other Leg 0.54829. As 0.83629: 325 :: 1: 388.6 As 0.83629: 325 :: 0.54829.: 213.07 CASE III. The Hypothenuse and one Leg given, to find the Angles and the other Leg. Fig. 41. |