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since the ratio of an arc to its sine is ultimately that of equality, and the sine of A+ dA may be considered as the same with the sine of A; it follows, that
I. d sin A= +cos AdA.
2. d cos A=–sinAdA.
13. Since, by No. 12. d sina-cosAdA, or the variation of the sine of an arc is proportional to its cosine; it follows that, near the termination of the quadrant, the slightest alteration in the value of a sine would occasion a material change in the
, or the
arc itself. Again, from the same Note, d tanA= dA co
variation of the tangent is inversely as the square of the cosine, and must therefore increase with extreme rapidity as the arc approaches to a quadrant.
14. It is convenient to reduce the solution of triangles to algebraic formulae. Let a, b and c denote the sides of any plane triangle, and A, B, and C their opposite angles. The various relations which connect these quantities may all be derived from the application of Prop. 11. bo-Eco–a’.
2. But, since (art. 16. NO. 3.) sino A*=#(1—cos A), it follows, by substitution, that in A-o-o-o-o: a’—(b–c)*_(a+b-c)(a—b-H c)
1. – 4bc
, and therefore, s denoting the
4. The second expression being now divided by the third,
gives tan #A* =olo, corresponding to Prop. 12.
These are the formulae wanted for the solution of the first case of oblique-angled triangles. To obtain the rest, another transformation is required.
sin A = 2T. For the same reason, sinE = #, and hence
6. Again, by composition, sin A-F sinBTao
, and therefore,
The preceding formulae will solve all the cases in plane trigonometry; but, by certain modifications, they may be sometimes better adapted for logarithmic calculation.
angle, its sine is the same as that of A+B, and consequently
-: aginAtoosasino). By a similar transformation, sin A
12. From art. 5, c=a
If the angle A be assumed equal to 90°, the preceding formulae will become restricted to the solution of right-angled triangles.
14. From art. 1., cosA= o- *::= whence, a”=b”-- co, which expresses the radical property of the right-angled triangle. #= +. and consequently sin B =+.
which corresponds with Prop. 7.
15. From art. 5.,
* - 16. Again, from the same article, _b__sinh sinE
Besides the regular cases in the solution of triangles, other combinations of a more intricate kind sometimes occur in practice. It will suffice here to notioe the most remarkable of these varieties. 20. Thus, suppose a side, with its opposite angle and the sum or difference of the containing sides, were given, to de