TRIGONOMETRY. 
05 
Sin 0° zz 0, meaning that 0 is the limit of size from 
which the sine begins, the revolving radius setting out 
from a. 
Sin 90° — R. 
Sin 180° — 0, meaning that 0 is the limit to which the 
sine approaches, the revolving radius coming to a'. 
Again, on the same supposition, it appears that the cosine 
cg, beginning from equality with the radius (when CD 
coincides with ca), will continually diminish, as at d, till cd 
reaches b, where it will vanish ; thenceforward it will con¬ 
tinually augment till it again becomes equal to the radius, 
when cd coincides with ca'. Hence at the same remark¬ 
able points a, b, a', the magnitude of the cosine will be 
thus expressed algebraically: 
Cos 0° zz R, meaning that the radius is the limit from 
which the cosine begins, the revolving radius setting out 
from a. 
Cos 90° — 0, meaning that 0 is the limit to which the 
cosine approaches, the revolving radius coming to b. 
Cos 180 — — R, meaning that the left-hand radius is 
the limit to which the cosine approaches, the revolving radius 
coming to a'. 
Finally, on the same supposition it appears that the tangent 
A li beginning from nothing (when cd coincides with ca), 
will continually augment, as at d', and become at length in¬ 
definitely great, when cd is at the point b ; for then the 
geometrical tangent ah being parallel to one side, cb, of the 
angle acb, the portion of it intercepted between both sides 
will be immeasurably great. When the revolving radius 
gets beyond b, as at d", then the intercepted portion 
above mentioned will be ah' contained between the sides 
ca and cd", the latter been produced backwards through 
c, so as to meet the geometrical tangent at h'. When 
the radius comes to a', the tangent again vanishes. Hence, 
at the same remarkable points a, b, a', the magnitude 
of the tangent will be thus expressed algebraically: 
Tan 0“ — 0, meaning that 0 is the limit from which 
the tangent begins, the revolving radius setting out from a. 
Tan 90° — oo , meaning that a quantity indefinitely great 
is that to which the tangent approaches, the revolving radius 
coming to b. 
Tan 180° — 0, meaning that 0 is the limit to which the 
tangent approaches, the revolving radius coming to a'. 
The preceding formulae may be registered thus: 
R R 
tan 45° — r sin 45° zz - >—= cos 45° zz . — cos 60° zz 
V 2 V 2 
sin 60° ~ v'JL.r sin 30°zz cos 30° zz • R 
2 2 2 a 
sin 0° zz 0, cos 0° zz R, tan 0° zz 0 sin 90° zz R, cos 90° 
zz 0, tan 90° zz oo, sin 180° zz 0, cos 180° zz — R, tan 
180° zz 0. 
From the above table we have a ready means of deter¬ 
mining the magnitude of such angles as have any of the 
sines, cosines, or tangents, there registered. For example, 
if it be required to find the magnitude of the acute angle 
(fig. 10.), bac, upon being given that the intercept ab, be¬ 
tween the perpendicular from one of its sides on the other, 
is half the distance ca; this angle is plainly equal to one- 
third of two right angles. Because, taking in the goniometric 
cirele, a cosine cg (fig. 9.), equal half the radius, the triangle 
gcd will be equiangular to bac (99 and 101 Geometry) in¬ 
asmuch as the angles at g and b are equal, and cg : cd : : 
ba : ac. Consequently the angle bac zz the angle gcd ; but 
the angle gcd is zz 60°, because its cosine zz , and there¬ 
fore also the angle bac zz 60°. 
But these formulas would suffice to determine the mag¬ 
nitude of but very few angles. It therefore becomes an 
object to construct a table in which the sines, cosines, &c„ 
of many different angles shall be registered; so that what¬ 
ever be the ratio of the three lines ab, ac, cb, amongst each 
other, we may find it in the table opposite some angle which 
will therefore be equal to the angle bac, whose magnitude is 
thereby determined. 
If in the formula (9) r X cos 2a zz r ‘ 2 — 2 sin 8 a, we 
zz r 2 —2 sin 1 15°, 
. •. 2 si id 15° 
• Ten . / R 2 R 2 \/3 
sm 15° zz \ / — — — 
V 2 4 
We have thus found the value of the sine of an angle zz 
15°; and by help of the same formula, we may find, in the 
same manner, the value of the sines of angles equal respec¬ 
tively to 7° 30' 3° 45', &c., &c., each successive angle being 
half the preceding. Likewise, inasmuch as the sine of an 
angle is equal to the cosine of its complement, the values 
thus found for the sines of angles equal respectively to 15°, 
7° 30', 3° 45', &c. &c., will be the values for the cosines of 
angles equal respectively to 75°, S2° 30’, 86° 15', &c. &c., 
which are the complements of the former. We might there¬ 
fore register these two series of sines and cosines, which would 
enable us to determine the magnitude of several angles. By 
a similar process we might compute the values of the sines, 
cosines, tangents, &c., of innumerable angles, and these 
values being registered, serve likewise to the above purpose. 
Now we may facilitate very much our operations by using 
a certain number to express a given radius, and using num¬ 
bers bearing relation therewith, to express the sine cosine, 
&c. Thus substituting 1 for r, we may change the pre¬ 
ceding formulae thus:— 
zz 0, cos 0° zz 1, tan 0° zz 0, sin 90° zz 1, cos 9d° zz 0, 
tan 90° zz oo, sin 180° zz 0, cos 180° zz — 1, tan ISO 0 
zz 0. 
The s n cs, cosnes, &c., heretofore spoken of, are now 
seldom registered in mathematical tables; instead of them 
we register their logarithms, which is found to be more con¬ 
venient in practice. The logarithms of the sines, cosines, 
&c. of angles, are called the logarithmic sines, cosines, 
&c., of these angles; and being put opposite their corre¬ 
sponding angles in the tables, are used instead of the natural 
sines, &c. 
We may calculate the logarithms of the natural sines. 
See., from these sines themselves; but for this purpose 
it is found most eligible to take the numeral radius zz 
10,000,000,000, or 10'°, instead of 1, and to consider the 
natural sines, &c., as made up of submultiple parts of this 
numeral radius, in the same manner as we before con¬ 
sidered them as made up of submultiple parts of 1. In fact, 
10’°, or any other number, may be looked upon as the unit 
or whole represented by 1. 
16. Let abc (fig. 11.) be any triangle, of which the three 
sides are represented severally by a, b, c, and the three cor¬ 
responding opposite angles by a, b, c. Then 
a : c :: sin a : sin c, 
and a : b :: sin a : sin b, 
and b : c :: sin b : sin c. 
Now represent the perpendicular from the vertex of any 
angle b on the opposite side by y. Then 
V 
~ zz sin 
c 
A, 
y 
- - zz sin c. 
a 
, *. sin 
