T ft I
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T R I
t£°. In every Triangle, as wel! plane as fpherical, the Sines of the Sides ? are proportional to the Sines of the oppo* fite Angles.
1 6°. In every plane 'Triangle, as the Sum of two Sides is to their Difference, fo is the Tangent of half the Sum of the oppofite Angles, to the Tangent of half their Difference. See Tangent.
!7°. If a Perpendicular he let fall upon the Bafe of an oblique angled 'Triangle $ the Difference of the Squares of the Sides is equal to double the Rectangle under the Bafe, and the Diftance of the Perpendicular from the Middle of the Bafe.
i8°- The Sides of a Triangle are cut proportionably, by a Line drawn parallel to the Bafe.
19 . A whole Triangle, is to a Triangle cut off by a right Line, as the Rectangle under the cut Sides, is to the Rectan- gle of the other two Sides.
2o°. In a right lined Triangle, a Line drawn from the right Angle at the Top perpendicular to the Hypothenufe, divides the Triangle into two other right lined Triangles, which are fimilar to the firft Triangle, and to one another.
2i°. In every right angled Triangle, the Square of the Hypothenufe is equal to the Sum of the Squares of the other two Sides. See Hypothenuse.
22 Q . If any Ancle of a Triangle be biffe&ed, the biffecV ing Line will divide the oppofite Side, in the fame Propor- tion as the Legs of the Angle are to one another. See Bissection.
-2 3°. If the vertical Angle of any Triangle be biffecled, the Difference of the Rectangles, made by the Sides and the Segments of the Bafe, is equal to the Square of the Line that biftects the Angle.
24 . If a right Line BE, Tig. 78. biffed an Angle ABC of a Triangle, the Square of the faid Line BE = AB-f-BC — A E -j- E C. Newt. Arith. Univerf.
'Properties of Spherical Triangles. See Spherical Tri- angle.
To divide a Triangle into any given Number of equay 'Parts: Divide the Bafe CD (Fig. 77.) into as many equal Parts as the Figure is to be divided into 5 and draw the Lines A 1, A 2, &c.
Triangle, in Trigonometry. — TheSolution or Analyfis of 'Triangles, is the Bufinefs of Trigonometry. See Trico- nometry.
The feveral Cafes thereof are reducible to the following Problems.
Solution of plane Tr i an g les.
i°. Two Angles A and C (Tab. Trigonometry, Fig. 27J 'being given, together with the Side A B oppofite to one of them -, to find the Side B C oppofite to the other.
The Rule or Canon is this : As the Sine of the Angle C, is to the given Side A B, oppofite to the fame 5 fo is the Sine of the other Angle A, to the Side requir'd. The Side BC, therefore, is commodioufly found by the Logarithms, from the Rule for finding a fourth Proportional to 3 Numbers given. See Logarithm.
For an Example: Suppofc C=48° 35'* ^ = 57* 28', A B— 74. The Operation will ftand thus :
Log. of Sine of C 9.8750142
Log. of AB 1.8692317
Log. of Sine of A 9.9258681
Sum of Loe . of A B ?
and of Sine of A $ M-79SW8
Log. ofBC 1.9200956. The Number corre-
fponding to which, in the Table of Logarithms, is 83, the Quantity of the Side fought.
2 . TwoSides A B and B C, together with the Angle C, op- pofite to one of them given ; to find the other Angle A and B.
The Rule is this : As one Side A B is to the Sine of the given Angle oppofite thereto C ; fo is the other Side B C, to the Sine of the Angle requir'd oppofite thereto.
S.gr. SuppofeAB = 94, BC = 6o, C = 7 ° 15'.
Log. of A B 1.973 "79 Lo?. of Sine of C 9-9/88i75 Log. of B C 1.8388491
Sum of Log. of Sine? ^ g
of C and or B ^ J
Log. of Sine of* A 9.9444387. The Number corre-
fponding to which, in the Table of Logarithms, is6l° 37'. Now the given Angle C being 72° 1 3', the Sum of the two; 133 52'fubftraaed from 180, the Sum of the three, gives 46 8' for the other Angle fought B.
In like Manner, fuppofe, in a right angled Triangle, (Fig. 28O that befide the right Angle, A, is given the Hypothenufe
B C, 49, and the Cathetus A 0,36, to find the Angle B ; Theft will the Operation ftand thus :
Log. ofBC 1. 690 1 96 1
Log. of whole Sine 10.0000000 Log, of A C 1.556302?
Log. of Sine of B 9.8661064. The correfponding Number to which, in the Table of Logarithms, is47° 16' 5 confequently, 0=42° 44'.
3 . Two Sides B A and AC, together with the included Angle A being given ; to find the two remaining Angles,
1. If the Triangle A B C be rectangular ; take one of the Sides including the Right Angle, as A B, for Radius ; then will C A be the Tangent of the oppofite Angle B: The Rule then is;
As one Leg A B, is to the other A C ; fo is the whole Sine to the Tangent of the Angle B.
E.gr. Suppofe B A 79, and A C 74 ; Log, of B A 18976271
Log. of AC 17323938
Log. of whole Sine ioooocooo
Log. of Tang, of B 98347667 ; The correfponding Num- ber to which, in the Table of Logarithms, is 34 21'; con- fequently the Angle C is 55 39'.
II. If the Triangle be oblique, the Rule is ; AstheSum of the given Sides A B and AC,F«j. 29. is to this ^Difference ; fois the Tangent of Half the Sum of the fmght Angles C and B, to the Tangent of Half the difference. Adding, therefore, the Half Difference, to the Half Sum ; the Aggregate will be the greater Angle C ; and fubtrafting rhe Half Difference from the Half Sum ; the Remainder is the Ids Angle B.
B.gr. Suppofe AB= 75, AC 58, A 108 24' ; Then will
AB75AB75 A+B + C 179° 60' AC 58 ACj8 A 10S 24
Sum 133 Differ. 17 B-fC
B + C 71 36
Log. ofAB-!-AC 2.1238516 Log.ofAB— AC 1.2304489 Log.of TangJB+C 9 8580694
Sum of Log. 12.0885183
L. of Tang . C— B S.9646667. The correfponding Num- ber to which is?^ 16' B+C = 3S° 48' B + C = 35 ° 4 g'
C-B = 5I6
C-B =
5= 16'
C =
41 4
30 32
4° The time Sides, A B, B C and C A (Fig. 30.) ham. given, to find the Angle A, B and C.
From the Vertex of the Angle A, with the Extent of the lead Side A B, defcribe a Circle : Then will C D be the Sum of the Legs A C and A B ; and C F their Difference. The Rule then is,
As the Bafe B C is to the Sum of the Legs CD • fo is the Difference of the Legs C F, to the Segment of the Bafe C G. — This Segment, thus f6und, being fubtraaed From the Bafe C B, the Remainder is rhe Chord G B. Then from A to the Chord G B let fall the Perpendicular A E - thus will BE = EG=i-GB. '
Thus, in a reftangled Triangle, A E B, the Sides A B and B E being given ; Or, in an obliquangled Triangle, ACE the Sides AC and C E being given : the Angles B and A are found.
JB.gr. Suppofe A B = 36, AC = 45, B C = 40.
AC = 4 5 AB 36
AC = 4 5 AB = 3 6
AC+AB=8i FC= 9
Log. of BC= 1. 6020600
Log. of AC-f AB 1.9084850
Log, of F C= o, 9542425
1 of Log. = 2. 8627275
. Log. of C G =,. 2606675 J The correfponding Number to which, in the lables, u ig. k e
BC = 4ooo
