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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.

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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

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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

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, and therefore, s denoting the

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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.

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sin A = 2T. For the same reason, sinE = #, and hence

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6. Again, by composition, sin A-F sinBTao

, and therefore,

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The preceding formulae will solve all the cases in plane trigonometry; but, by certain modifications, they may be sometimes better adapted for logarithmic calculation.

2 C.

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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

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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

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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

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The relation of the sides and angles of a triangle might also be in some cases conveniently expressed by a converging se- b sinb sinE sinb ties. Thus +=ji=sin(sic) =jobic” and consequently b sinB cosc + b cos B sin C = a sin B, or b sinC sinE

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