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OMETRY by which the sides and Angles of Triangles are measured ; whereby three things being giv. en, either all Sides or Sides and Angles, a fourth may be found; either by measuring with a Scale and Dividers, according to the PROBLEMS in GEOMETRY, or more accurately by calculation with Logarithms, or with Natural Sines.
TRIGONOMETRY is divided into two parts, Rectan, gular and Oblique-angular.
RECTANGULAR TRIGONOMETRY. This is founded on the following methods of applying a Triangle to a Circle.
PROPOSITION I. In every Right Angled Triangle, as ABC, PLATE II. Figure 44. it is plain from PLATE I. Fig. 7. compared with the Geometrical Definitions to which that Figure refers, that if the Hypothenuse AC be made Radius, and with it an Arch of a Circle be described from each end, BC will be the Sine of the Angle at A, and AB the Sine of the Angle at C; that is, the Legs will be Sines of their opposite Angles.
PROPOSITION II. If one Leg, AB, Fig. 4.5.
be described, then BC, the other Leg, will be the Tangent and AC the Secant of the Angle at A; and if BC be made Radius, and an Arch be described with it on the Point C, then AB will be the Tangent and AC the Secant of the Angle at C; that is, if one Leg be made Radius the other Leg will be a Tangent of its opposite Angle, and the Hypothenuse a Secant of the same Angle.
Thus, as different Sides are made Radius, the other Sides acquire different names, which are either Sines, Tangents or Secants.
As the sides and Angles of Triangles bear a certain proportion to each other, two Sides and one Angle, or one Side and two Angles being given, the other Sides or Angles may be found by instituting Proportions, according to the following Rules.
RULE I. To find a Side either of the Sides may be made Radius, then institute the following Proportion : As the name of the Side given, which will be either
Radius, Sine, Tangent or Secant; Is to the length of the Side given ; So is the name of the Side required, which also will
be either Radius, Sine, Tangent or Secant ; To the length of the Side required.
RULE II. To find an Angle one of the given Sides must be made Radius, then institute the following Proportion :
As the length of the given Side made Radius;
Secant, Having instituted the Proportion, look the corresponding Logarithms, in the Logarithms for Numbers for the length of the Sides, and in the Table of Artificial Sines, Tangents and Secants, for the Logarithmic Sine, Tangent or Secant.
Having found the Logarithms of the three given Terms, add together the Log. of the second and third
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 Loga,
rithmic Tangent of 45°. The preceding PROPOSITIONS and Ryles being duly attended to, the solution of the following Cases of Rectangular Trigonometry will be
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.
To find the Leg BC. As Radius 10.00000 As Radius
10.00000 : Hyp. AC, 25
: Hyp. AC, 25
1.39794 :: 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. To find the Leg BC. As Secant CAB, 35° 30' As Secant CAB, 35° SC' : Hyp. AC, 25
: Hyp. AC, 25 :: Radius
:: Tangent CAB, 35° 30' '
Making the Leg BC Radius, the Proportions will
To find the Leg AB. To find the Leg BC. As Secant ACB, 54° 30' As Secant ACB, 54° 30' : Hyp. AC, 25
: Hyp. AC, 25 :: Tangent ACB, 54° 30' :: Radius : Leg AR, 20.35
: Leg. BC, 14.52 · 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
bcing 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.
Making the given Leg Radius, the Proportions will
To find the Hypothenuse. As Radius,
10.00000 : Leg AB, 325
2.51188 :: Sec. CAB, 33° 13' 10.07765
To find the Leg BC.
10.00000 : Leg AB, 325
2.51188 :: Tan. CAB, 33° 15' 9.81666
: Hyp. 383.6
: Leg BC, 213.1
Note. Reject the first figure, which is the same as
subtracting Radius, and seek the numbers corres
ponding to the other figures. Making the Leg BC Radiuş, the Proportions will be; To find the Hypothenuse. To find the Leg BC. As Tang. ACB, 56° 45' As Tang. ACB, 56° 45' : Leg AB, 325
: Leg AB, 325 :: Sec. ACB, 56° 45' :: Radius