Plane trigonometry, by S.L. Loney.

54

TRIGONOMETRY.

In the fourth quadrant as the angle increases from
270~ to 360~, the sine increases from -- to -, i.e. from
a a
-1 to 0.
55. Cosine. In the first quadrant the cosine, which
OM K a 0.
is equal to- -- decreases from - to, i.e. from 1 to 0.
a a a
0 -a
In the second quadrant it decreases from - to, i.e.
a a
from 0 to - 1.
In the third quadrant it increases from-a to -, i.e.
ca a
from - 1 to 0.
o a
In the fourth quadrant it increases from - to -,.e.
a a
from 0 to 1.
56. Tangent. In the first quadrant MJ/,P, increases
M1P,
from 0 to a and OM, decreases from a to 0, so that OMcontinually increases (for its numerator continually increases and its numerator continually decreases).
When OP1 coincides with OA, the tangent is 0; when
the revolving line has turned through an angle which is
slightly less than a right angle, so that OP1 nearly
coincides with OB, then MAP1 is very nearly equal to
a and OM1 is very small. The ratio --- is therefore very
O1V
large, and the nearer OP1 gets to OB the larger does the
ratio become, so that, by taking the revolving line near
enough to OB, we can make the tangent as large as we
please. This is expressed by saying that when the angle
is equal to 90~ its tangent is infinite.