Plane trigonometry, by S.L. Loney.

CHANGES IN THE TRIGONOMETRICAL RATIOS. 55

The symbol co is used to denote an infinitely great
quantity.
Hence in the first quadrant the tangent increases from
0 to so.
In the second quadrant when the revolving line has
described an angle AOP2 slightly greater than a right
angle, ]MJP2 is very nearly equal to a and OM2 is very
small and negative, so that the corresponding tangent is
very large and negative.
Also, as the revolving line turns from OB to OA', M1PM
decreases from a to 0 and OM1 is negative and decreases
from 0 to -- a, so that when the revolving line coincides
with OA' the tangent is zero.
Hence in the second quadrant the tangent increases
from - oo to O.
In the third quadrant both MiP3 and 0M1 are negative,
and hence their ratio is positive. Also, when the revolving
line coincides with OB', the tangent is infinite., Hence in the third quadrant the tangent increases
from 0 to oo.
In the fourth quadrant M4P is negative and OM is
positive, so that their ratio is negative. Also, as the
revolving line passes through OB' the tangent changes
from - co to - oc [just as in passing through OB].
Hence in the fourth quadrant the tangent increases
from - oo to 0.
57. Cotangent. When the revolving line coincides
with OA, M1P1 is very small and OM1 is very nearly
0M,1
equal to a, so that the cotangent, i.e. the ratio M P', is
infinite to start with. Also, as the revolving line rotates