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XI. A proposed Improvement in the Solution of a Case in Plane 

 Trigonometry. By WILLIAM WALLACE, F. R. S. Edin. 

 Professor of Mathematics in the University of Edinburgh. 



(Read Nov. 23. 1823J 



4 ' 



1. IN the present state of mathematical science, cultivated as 

 it has been, with assiduity, during the two preceding centuries, 

 it can hardly be expected that any considerable improvement re- 

 mains to be made in Plane Trigonometry, one of its most ele- 

 mentary theories. There is, however, one case in the resolution 

 of oblique-angled triangles, which appears to me to admit of a 

 solution somewhat more simple and convenient than those which 

 are commonly known ; it is that in which two sides and the in- 

 cluded angle are given to find the third side. 



2. The usual way of proceeding, is to find half the sum and 

 half the difference of the angles opposite the given sides, and 

 from these the angles themselves ; the third side may then be 

 found in two ways, from the principle, that the sides are to one 

 another as the sines of the opposite angles. Instead of this, I 

 propose, that having found half the sum and half the difference 

 of the angles in the usual way, the remaining side shall be found 

 by either of these two formulae : 



Let the sides of a triangle be a, b, c, 

 and the opposite angles A, B, C, 



Theorem I. 



cos i (A B) : cos i (A + B) : : a + b : c. 



Theorem II. 



sin (A B) : sin HA + B) : : a b : c . 



