424 



Strong'^s Problems. 



MATHEMATICS. 



^1 join X 



Art. XX. An improved Method of obtaining the Formula 

 for the Sines and Cosines of the Sum and Difference of 

 tzooJlrcs, by Professor. Strong, of Hamilton College. 



In the circle ABCD let AB 

 and BC denote any two arcs 

 contiguous to each other. Draw 

 their limiting diameters Aa, 

 Cc ; their sines Bx, By ; and 

 y. Then will xy = sine 

 of (AB + BC): for if upon OB 

 as a diameter we describe a 

 circle, it will manifestly pass 

 through the points x and y, 

 (since the angles OrrB, OyB 

 are right, see Euc. 31. 3.) therefore OxBy is a quadrilateral 

 inscribed in a circle described on OB as a diameter, and the 

 angle yOx at the circumference stands upon an arc whose 

 chord is xy. Again, if from a we draw ad perpendicular to 

 Cc, it will be the sine of the arc ac (— AB+BC). If now 

 we describe a circle on aO as diameter, it will pass through d, 

 (see Euc. 31. 3.) therefore adis the chord of an arc on which 

 the angle aOc stands in the circle described on aO. But in 

 equal circles the chords of arcs on which equal angles at the 

 centres or circumferences stand are equal ; (see Euc. 26. and 

 29. 3.) hence xy = ad = sin(AB -1- BC). Now sine OxBy is 

 a quadrilateral inscribed in the circle described on OB as dia- 

 meter, we shall have (Eue. D. 6.) OB'xy = Bx-Oy -}- B?/'Ox= 

 sinAB- cosCB -{- sinCB- cosAB. If OB be denoted by r, we 

 shall have xy, or sin(AB + BC)= 



sinAB- cosCB -{- sinCB- cosAB 



If AB = A, BC = B, and the radius r = 1, sin(A -1- B) = sin 



