Solution of the Cases of Oblique Angled Triangles. i GENERAL PROPOSITION. In an oblique angled triangle, of the three sides and three angles, any three being given, the other three may be found, except when the three angles are given; in which case the ratios of the sides are only given, being the same with the ratios of the sines of the angles opposite to them. 2 AB, AC, and The angles B, two sides and A and C. an angle opposite to one of them. S, C: S, A :: AB: BC, Fig. 16. 17. and also S, C: S, B:: AB : AC. (2.); AC: AB:: S, B: S, C. (2.) This case admits of two solutions; for C may be greater or less than a qua!drant. (Cor. to def. 4.) 3 AB, AC, and The angles AB+AC: AB-AC :: T, A, two sides, B and C and the in cluded angle. 2 ACxCB: ACq+CBq -ABq:: R: CoS, C. If ABq+CBq be greater than ABq. FIG. 16. 2 AC × CB: ABq-ACq -CBq:: R: CoS, C. If 4 AB, BC, CA, A, B, C, the ABq be greater than ACq+ the three sides. three angles. CBq. FIG. 17. (4.) Otherwise, Let AB+BC+AC=2P. PxP-AB: P-AC X P-BC:: Rq: Tq, C, and hence C is known. (5.) Otherwise, Let AD be perpendicular to BC. 1. If ABq be less than ACq+CBq. FIG. 16. BC: BA+AC:: BAAC: BD-DC, and BO the sum of BD, DC is given; therefore each of them is given. (7.) 2. If ABq be greater than ACq+CBq. FIG. 17. BC: BA+AC:: BA-AC:BD +DC; and BC the difference of BD, DC, is given, therefore each of them is given. (7.) And CA: CD:: R: Co S, C. (1.) and C being found, A and B are found by case 2 or 3. CONSTRUCTIONS OF THE TRIGONOMETRICAL CANON. re A Trigonometrical Canon is a Table, which, beginning from one second or one minute, orderly expresses the lengths that every sine, tangent, and and secant have, in spect of the radius, which is supposed unity; and is conceived to be divided into 10000000 or more decimal parts. And so the sine, tangent, or secant of an arc, may be had by help of this table; and, contrariwise, a sine, tangent, or secant being given, we may find the arc it expresses. Take notice, that in the following tract, R signifies the radius, S a sine, Cos. a cosine, Ta tangent, and Cot. a cotangent; also ACq signifies the square of the right line AC; and the marks or characters, +,,,, ::, and ✓, are severally used to signify addition, subtraction, equality, proportionality, and the extraction of the square root. Again, when a line is drawn over the sum or difference of two quantities, then that sum or difference is to be considered as one quantity. Constructions of the Trigonometrical Canon. PROP. I. THEOR. THE two sides of any right angled triangle being given, the other side is also given. For (by 47. 1.) ACq=ABq+BCq, and ACq-BCq= Fig. 28. ABq, and interchangeably ACq-ABq=BCq. Whence, by the extraction of the square root, there is given AC= ✓ABq + BCq; and AB = ACq-BCq; and BC ✓ACq-ABq. PROP. II. PROB. THE sine DE of the arc BD, and the radius CD, Fig. 29. being given, to find the cosine DF. |