18 MATHEMATICS. 
Instead of the angles, certain quantities are emp.oyed whose value 
depends on that of the angle, and which are called the trigonometrical 
functions. The most important of these are the sine, cosine, tangent, and 
cotangent. The explanation of these may be best made in a right angled 
triangle (fig. 105). Here the side opposite an acute angle, as abc or m, 
ac 
divided by the hypothenuse, as ~, is called the sine of that angle, likewise 
the cosine of the other acute angle, bac or n; 2, the side opposite an acute 
ac 
angle, abc, divided by the other short side, as 7 is the tangent of that 
angle, and likewise the cotangent of the other acute angle, bac. Conse- 
quently — 
4 ac 
Sit. wees 
ab 
; be 
Sin. 2#—cos. m——— 
ab 
ac 
Tang. m—cot.n =__ 
a 
be 
Wane cot. ie 
ac 
Consequently, in similar right angled triangles of different size, the sines, 
cosines, &c., of the homologous or corresponding angles will be equal. If 
the hypothenuse of the right angled triangle be taken as unity, then the 
side opposite an acute angle may be taken as the sine of that angle and the 
cosine of the other. How far the sine, cosine, &c., of an angle varies with 
its size, may be seen in fig. 106. Here abe is a quadrant whose radius is 
taken as unity. Consequently, de = sin. dbe; fg = sin. fbc; be = cos. 
dbc; bg = cos. fbg; whence it follows that the sine of an (acute) angle is 
greater, and the cosine less, as the angle is greater. Consequently, the 
tangents likewise increase, and the cotangents diminish as the (acute) 
angle increases. The sines and cosines of (acute) angles are evidently 
always fractions, while the tangent and cotangent of 45°=— 1; tangents of 
more than 45° are greater than unity, and as the angle approaches 90°, 
they become very great, tang. 90°= infinity; the same is the case with 
the cotangents of angles less than 45° and approaching 0. 
The sines, cosines, tangents, and cotangents of all acute angles, have 
been calculated and arranged in tables called trigonometrical tables, which 
are indispensable in all trigonometrical calculations. The ordinary tables, 
however, do not contain the sines, cosines, &c., themselves, but their 
logarithms, as these are more readily employed in calculations. 
From the preceding exp anations may be readily derived rules for solving 
all possible cases of right angled triangles. For acute angled triangles, the 
following two propositions are of the greatest importance :—1l, any two 
sides of a triangle are to each other as the sines of their opposite angles 
(pl. 3, figs. 107, 108). In fig. 107, the triangle abc is divided into two right 
angled triangles, abd and acd, by the perpendicular let fall from @ on be. 
18 
