96 
TRIGONOMETRY. 
sin A : sin c 
c a 47 
cy 
la like manner it is shown that s/m a : s/m b r: a : 5, and 
sin b : sin c :: 6 : c, by drawing perpendiculars from the 
vertices of the other angles c and a to the other sides 
c and a. q. e. d. 
17. Let abc (fig. 12), be a triangle, whose sides and angles 
are represented, as above, by a, b, c, and a, b, c. Then 
tan 
:: 0 + 6 = a ~ b ' 
Produce ba, which is not the greater of the two 
sides under consideration, until it be equal to the other bc ; 
join cd, and let fall on it the perpendicular be, which 
divides it equally at e. The angle bcd — bdc ; therefore, 
because “ a triangle that has two sides equal, has also 
its angles opposite the equal sides equal bdc — bca + 
a cd. But, by 95 Geom. baC — bdc 4- ac;d, .\ bag — 
BCA 4- 2aCD, .'. ACD — 
BAC-BCA A -C 
Likewise, 
2 ~~ 2 
as bac — bdc -f- acd, adding the angle bca to both, we 
get bac 4- bca — bdc 4- acd 4 - bca — 2bdc — 2bcd 
BAC + BCA A 4- C 
BCD = - 2 - =—2 — 
Again: ad — bd • 
.. . • , AD BC BA 
divided at f, af — — ~-g- 
ba — bc — ba ; .•. if ad be equally 
a — c , - 
— T) —; and 2 bf — 
2 ba 4 - 2 af — bd 4 ba — bc 4 ba, . \ bf — 
a 4 c 
BC 4- fi A — 
Finally : Joining fe, fe is parallel to ac, by 2d of 6 th 
of Euclid: consequently be : 3e : : bf : af, by the same; 
that is, representing the perpendiculars be and £e respectively 
by y and y’, as also the line eg by x, we have 
y ‘ y BF : AF, 
V y' a 4 c a — c 
.*. ~ : — :: bf : af :: — ’— : —-— :: a 4- c : a — c. 
xx 2 2 
But, tan bcd — tan —and — — tan acd ~ 
tan 
x 
A -C 
2 ’ 
tan 
hence 
A 4- c 
tan 
a — c 
: a 4 - c : a — c. 
In like manner it is shown that tan —: ^~ 2 ~ 
a b : a — b ; and that tan 
b + c b — c . 
~2 : ::-5 + c: 
b — C. Q. E. D. 
18. Let abc (fig. 13.) be a triangle, whose sides and angles 
are represented as above by a, b, c, and a, b, c. Then cos 
a X 2 bc — b 2 4- 
Represent the perpendicular bd by y, and dividing ac 
equally at e, represent the intercept ad by x. The inter¬ 
cept DC will be represented by(6— x) Then by Geom. Ill, 
t? —y 2 4- x 2 , and a 2 — y 2 -4 (b — x) 2 , 
.. ,*—s* = t fi—(b—xfr 
c? — a 2 = x 2 — ( b —x) 2 — — b 2 4 2bx, 
.-. 2 bx — b 2 4 - c 2 — a 2 , 
2bc X — — b 2 4- c 2 — a 2 , 
c 
, . x 
that is, as cm a z —, 
c 
cos A X 2 bc — b 2 4- c 2 — a 2 , 
or, in another form, 
b 2 4- c 2 ' 
COS A — — 
2 bc 
Q. E. D. 
Obs. We are now in a condition to proceed to the mea¬ 
surement of triangles, or Trigonometry, properly so called. 
In a triangle there are three sides and three angles: certain of 
these being given in magnitude, determine the triangle both 
in form and magnitude. For, 
1°. Two sides and the included angle being given, the tri¬ 
angle can be of but one form and magnitude, by 99 Geom. 
2°. Three sides being given, the triangle can be of but one 
form and magnitude, by 101 Geom. 
3°. Two angles and a side being given, the triangle can be 
of but one form and magnitude, by 101 Geom. 
3°. Two angles and a side being given, the triangle can be 
of but one form and magnitude, by 102 Geom. 
There are no other three of the above six parts which will 
absolutely and completely determine the triangle. For if 
the three angles be given, the triangle is determined indeed as 
to form, inasmuch as all triangles with three such angles must 
by 167 Geom. have their sides respectively proportional to each 
other, and therefore be similiar in figure. Thus, if figs. 14 14 
abc, abc , be two triangles having the three angles at A, b, c, 
equal respectively to those at a , b, c; then, ab : ab :: bc : 
bc, and ab : ab : : ac : ac, and bc : bc: : ac : ac; so that, 
having their angles equal and their sides proportional, these 
triangles are similar in form. But they are, or at least may 
be, very unequal in magnitude, and therefore a triangle is 
not determinable from such data, completely as to form and 
magnitude, but partially as to form alone. 
So also if two sides and an angle opposite one be given, 
the triangle may be determinable neither as to form nor mag¬ 
nitude. 
Two sides and an angle opposite one being given, the 
triangle can be of but of one, or a second, form and mag¬ 
nitude. 
91. Let a, b, c, (fig. 16.) be respectively the three sides of 
the right-angled triangle ; a, b, c, the corresponding oppo¬ 
site angles, a being the right one. Then, b, c, a, being given 
or known, a, b, c, may be determined. 
- - ■ - l) Q 
Dem. For a — >J b' 1 X c 2 ; tan b — ; and tan c — -r 
Q.E.D. b 6 
Thus, suppose it were given that the two sides, b and c, 
about the right-angle of a right-angled triangle were respec¬ 
tively equal to 3 and 3-y/ 3 (feet, inches, or any other li¬ 
near unit); then, we should have 
b 2 — 3 2 r= 9, 
c 2 = (3,v/3 ) 2 r= 27, 
.-. a — x/9 4- 27 — 6 . 
Likewise j _ 
tan b — tan c — 3 
a/3 
Here we have determined the third side, a, to be 6 (feet, 
inches, or whatsoever was the linear unit chosen). The two 
acute angles are also determined; because to find them from 
having their tangents, we have only to consult a trigonome¬ 
trical table, and whatever angles in it have tangents of the 
values found, are equal to the angles computed. Thus, 
if 1 be the tabular radius, we shall see —~ registered as the 
^3 
tangent of 30°, and y'd as the tangent of 60°; for tan 
30° = 
sin 30° 
cos 30° 
and tan 60° ~ 
V3 
sin 60° _ 
cos 60° — V'S- 
Obs. 1. As the angles b and c are together equal to a 
right angle, having determined either, b, we may obtain the 
other immediately; for c — 90°— b. Thus, if we find that 
b is — 30°, c must be — 90° — 30° ~ 60°. 
Obs. 2. Instead of looking in a trigonometrical table for 
angles corresponding to the tangents so found, we more fre¬ 
quently take the logarithms of those values found for the tan¬ 
gents, and then consult a table. Thus, if the sides of the 
above triangle were b ~ 43, and c ~z 55, we should have 
(using the tabular radius r~ 10 10 ), 
b 
tan b — r. --, 
••• log. 
