Strong’s Problems. 
MATHEMATICS. 
« : — S208 
Arr. XX. ; An improved Method of obtaining the Formule 
for the Sines and Cosines of the Sum and difference of 
two Arcs, by Proressor Srrone, of Hamilton College. 
Iv the circle ABCD let AB 
and BC denote any two ares 
contiguous to eachother. Draw 
B their limiting diameters Aa, 
U4 Cc; their sines Br, By; and 
join z,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 2 and y, 
(since the angles OxB, OyB are right, see Euc. 31. 3.) there-_ 
rBy 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 Ce, it will be the sine of the are ac 
(=AB+BC.) If now we describe a circle on aO as diameter, 
it will pass through d,(see Euc. 31. 3.) therefore ad is the chord 
of anarc on which the angle aOc stands in the circle described 
onaQ. But in equal circles the chords of ares on which equal 
angles at the centres or circumferences stand are equal; (see 
Euc. 26. and 29. 3.) hence zy=ad=sin(AB+BC.) Now 
since OxBy is a quadrilateral inscribed in the circle described 
on OB as diameter, we shall have (Euc. D. 6.) OB-xy=Be- 
Oy+By-Ox=sinAB: cosCB+sinCB: cosAB. If OB be de- 
noted by r, we shall have xy, or sin (AB+BC)= 
sinAB- cosCB+sinCB- cosAB. 
r 
If AB=A, BC=B, and the radius r=1, sin(A-+B)=sin 
i 
A EN ee a ee 
