97 
TRIGONOMETRY. 
log. tan b — log. r + log. —, 
— log. r 4- log. b.— log. c. 
— log. 10 10 + log. 43. —log. 55. 
Sut in a logarithmic table whose base — 10, 
log. 10 l ® — 10 
log. 43 — 1,6334685 
log. 10'®4- log. 43 rr 11,6334685 
and, log. 55 — 1,7403627 
.•. log. 10 10 + log. 43 — log. 55 — 9,8931058, 
which logarithm corresponds to an angle — 38° 1' 8 ", . •. b 
— 38° 1'8", and .-.0 = 51° 58' 52". 
Again, _ 
a rr + c 2 ; 
therefore a may be found by extracting the square root of 
42° 4- 55 2 . It is easier, however, to find the value of a by 
logarithms, to which the preceding formula is not adapted, 
being unresolvable into factors, and thereby preventing us 
from substituting the sum of their two logarithms for the 
logarithm of their product. We must, therefore, obtain the 
value of a by means of some other formula, viz.; 
c 
cos B~r — . 
a 
Q. E. D. > 
Triangles which are not right-angled may be solved, that 
is, determined from certain data, with little less facility than 
the above. 
23. Let a, b, c, (fig. 17.) be respectively the three sides of 
the triangle, a, b, c, the opposite corresponding angles. Then, 
any two sides, as a, b, and the included angle c, being 
known, the third side c, and the remaining angles a, b, may 
be determined. 
Dem. (By 17.) 
¥Y~ l 
A -f-B 
2 
2 ) V 2 ) a -J- b ’ 
but as a 4- b 4- err 180°, a + b — 180°— c. 
— 90° ——. Consequently, subtracting half the given angle 
c from a right angle, we obtain 
a4-b 
and therefore can 
. •. a rr r --> 
COS B 
. •. log. a — log. r 4- log. 
— log. r 4- log, c — log. cos b, 
— 10-j- 1,7403627—log. cos 38° 1'8"; 
but, in the table of logarithmic cosines, 9,8964202 corre¬ 
sponds to an angle of 38° 1' 8", and therefore we have 
— log. a rr 10 4- 1,7403627 — 9,8964202, 
— 1,8439425; 
and, as 1,8439425 is the logarithm of the number 69, 81, 
&c., we find 
a rr 69, 81, &c. 
20. a , b, a, (fig. prec.) being given, c, B, c, may be de¬ 
termined. 
______ b 
Dem. For c — y'a 2 — b‘\ by 111 Geom.; and sin b 
b 
cos c rr — 
a » 
In logarithms, as 
— rr v'C a+b).(a — b) rr (a 4- P. {a—by ,. •. 
log. err log- (a 4- by 2 .(a — b)^ ~ log. (a 4-6)' 4- log. 
(a — b )s; log. e — \ log. {a-\-b)-\-\ log. (a —6). 
And, 
log. sin b zr log. b — log. a, 
log. cos c z: log. b — log. a. 
Q. E. 0. 
21. a, b, a, c, (fig. prec.) being given, b, c, may be 
determined. 
b c 
Dem. For, sin b rr—. b~ a .sin b ;and — rr cos b, 
. ■. c rr a . cos b. 
In logarithms, 
log. b rrlog. a 4- log. sin b, 
and, log. c rr log. a + log. cos b. 
Q. E. D. 
22. Representing the sides and angles as above, then 
b, a, b, c, being given, a, c, may be determined. 
Dem. For sin b rr—,. •. a —-; and — rr tan b, 
a sin b ’ c 
__ b 
tan b‘ 
In logarithms, 
log. a rr log. b — log. sin b, 
and, log. c rrlog. b — log. tan B. 
Vol. XXIV. No. 1630. 
find its tangent in the tables. Hence, as a and b are likewise 
A _ B 1 
given, we obtain the whole value of tan ——, from whence 
* 
A—.g 
we get the corresponding angle — ■ b y the tables. But 
from half the sum —, and half the difference —’ °f the 
two angles a and b, we can determine both separately; as 
we have only to add half the difference to half the sum for 
the greater, and to subtract half the difference from half the 
sum of the lesser. Thus, if c rr 40°, then- A - i~ — rr 90 ®—~ 
2 2 
— 70“; and if, from the above method of computation, we 
find, by the tables, that — ~ B rr 10, then 
4-^-T® — a rr 70® 4- 10“ rr 80®, 
and, — £=5- rr b rr 70® — 10® rr 60®. 
2 2 
Again: having thus computed A and b, we may find c 
For, (by 16,) 
c sin c sin c 
— rr ——, .•. c r: a . —,—• 
a sin a sin a 
from which formula for c its value may be determined, as a, 
sin c and sin a, are known quantities. 
In logarithms, 
log. c rr log. a 4- log. sin c — log. sin A. 
Q. E. D. 
24. Representing the sides and angles as above, then 
any two angles, as a, b, and a side, as a , opposite either of 
these angles, a, being given, the third angle, c, and the re¬ 
maining sides, b, c, may be found. 
Dem. The third angle, c, is found by subtracting the sum 
of a and b from 180°; for a 4- B + c rr 180°, .•. c rr 
180° — (A 4- b)- Likewise, by (16,) 
a 
and, — rr 
a 
Hence, b and c are found in terms either given as a, or 
easily computable from trigonometrical tables, as sin a, sin 
B, sin c. 
In logarithms, 
log. b rr log. a 4- log- sin b — log. sin a. 
and, log. c rr log. a 4- log. sin c — log. sin a. 
This, &c- 
25. For two angles being given the third is determin¬ 
able, as in the prec.; and the solution of this question will 
be thus reduced to the last, 
26. Representing the sides and angles as above, then 
a, b, c being given, a, b, c may be found, 
2 C Dem 
sin b 
. 6 zz a 
sin b 
sinA 9 * 
* sin A 
sin c 
sin c 
7 y • 
\ czz a 
• —:-, 
sin a 
sin a 
