TRIGONOMETRY. 19 
: ad : ad 
From the first we have sin. 1 tae ; from the second, sin. n—=-—— ; whence 
a ac 
sin. m:sin. 2:: 1 ac:ab. In ig. 108, where the triangle abc is 
ab ac 
obtuse angled, and the perpendicular let fall from c meets only the prolon- 
: : cd : cd , , 
gation of ab, we have sin. o=— and sin. n= — whence sin o: sin. n:: 
c 
bc: ac; so that the preceding proposition holds good also for obtuse angled 
triangles, if, instead of the sine of the obtuse angle, we take that of the 
angle which must be added to the obtuse angle, to make two right angles. 
2. The sum of two sides of a triangle is to the difference of these sides, as 
the tangent of half the sum of the angles lying opposite to them, is to the 
tangent of half their difference. In the triangie abe (fig. 109), we accord- 
ingly have ab+-ac: ab—ac::tang. $ (acb + abc): tang. 3 (acb—abc). In 
the figure, with the lesser of the two sides, ab and ac, namely ac, a semi- 
circle is described cutting ab and its prolongation in d and e, the chords 
ed and ce drawn, as also df parallel to ce. Then cdf and dce being right 
angles, we have be: bd, that is ab + ac:ab—ac::ce:df. But ce—cd tang. 
z, and df=cd tang. y; moreover, © = } cae = 3 (acb+abc) ; and y= x — 
n == %4(acb—abc), whence the preceding proposition immediately fol- 
lows. 
If we distinguish the angles of a triangle by A, B, C, and the sides 
opposite to each by a, b, c, we have the following formula for the solution 
of triangles. 
I.—For right angled triangles, when A is the right angle. 
1. Given the hypothenuse a, and a side 0; then sin. B= Be oa. 
a 
cos. B. 
2. Given the hypothenuse a, and an acute angle B; then b—a.sin. 
B; c=a.cos. B. 
3. Given the two sides 6 and c; then tang. peat th We 
c sin. B 
= € é 
~ cos. B 
4, Given the side }, and an acute angle B or C; then a= — a 
sin. 
= b ;c=b, cot. B=, tang.C. 
cos. C 
1].—For acute angled triangles. . 
1. Given a side, a, and two angles; then b= Ee, c= iS 
sin. A sin. A 
2. Given two sides, a, b, and an opposite angle, A; then sin. B-= 
bsin. A asin. _ 6sin. C 
—— ——<—— 
——_—_—_—_—_ =—— e 
3 
c= — : 
a sin. A sin. B 
19 
