Strong'^s Problems. 425 



A- cosB +sinB'cosA; which is the known formula for the 

 sine of the sum of two arcs, to the radius 1. 



Again, if through O we draw the diameter DE perpendicular 

 to Aa, then will DC be the complement of (AB -}- BC). Draw 

 Cp, the sine of DC = cos(AB -f BC). Through B draw the 

 diameter B6 ; from b, draw the sines bz, br, of the arcs fee, 

 iE respectively, and join z, r. Then by describing two cir- 

 cles, one on 60 as diameter, the other on OC, it may be proved 

 as before that the circle described on iO passes through the 

 points z and r, and that the circle described on CO passes 

 through p : and hence, by the same reasoning as before, zr = 

 Cp = cos(AB -}- BC). Now Obzr being a quadrilateral in- 

 scribed in the circle described on 60, we have (by the prop, 

 before cited) bO- zr -{- Or- bz = br. Oz ; and hence bO- zr = 

 bvOz — Or' bz. Bui 6r = sine arc 6E = sine arc BD ; and 

 since BD is the complement of AB, br = cosAB. In like 

 manner Oz = cosBC, Or = sinAB, and bz = sinBC ; hence 

 by substitution, 60* zr = cosAB* cosBC — sinAB* sinBC. By 

 using the same notation as before, we have co3(A + B) 



cosA- cosB — sinA. sinB „, . , t. . . . 



= =(if r = 1) cosA- cosB — smA-sm 



B, which is the known formula for the cosine of the sum of 

 two arcs. 



The same construction will answer for the two remaining 

 cases : for if we suppose that 6E and be are two arcs, then will 

 cE be their difference, and zr the sine of cE, as proved above ; 



hence zr (= sin(6E — 6c))= — '- '- But br =: sin 



fcE, and Or = its cosine ; and bz = sine be, and Oz = its cos. 



hence if 6E be denoted by a, be by 6, and 06 as before, then 



.„ . ^ ,, sina- cos6 — sin6. cosa ... ^. . , 



willsm(a— 6) = =n (ifr = 1) sina* cos6— 



sin6.cosa'— . Again, AB+BC is the complement of DC 



or cE ; hence by the first part of the above investigation, 



xy = sin(AB -\- BC) = coscE ; but xy or sin(A-|-B) = cos 



, ,. sinA* cosB + sinB" cosA , . . .„ 



(a— 6) = ■ . and as smA or AB 



= cosBD = cos6E, Or = cosA or AB = sinBD := sin6E, By 



