Plane trigonometry, by S.L. Loney.

SIN 0 < 0 < TAN 0.

261

The straight line PP' is less than the arc PAP', so
that NP is < arc PA.
We shall assume that the arc PAP' is less than the
sum of PT and TP', so that arc PA < PT.
Hence NP, the arc AP, and PT are in ascending
order of magnitude.
Tro NP areAP.PT
Therefore NP, aOP and -p are in ascending
OP ' OP OP
order of magnitude.
NP
But P = sin AOP = sin 0,
arc P = number of radians in z AOP = 0 (Art. 21),
PT
and = tan POT = tan AOP = tan 0.
OP
Hence sin 0, 0, and tan 0 are in ascending order of
magnitude, provided that
0<
228. Since sin 0< 0< tan 0, we have, by dividing
each by the positive quantity sin 0,
0 1
1 sin 0< cos
Hence i.-n always lies between 1 and --
sin 0 a cos 0
This holds however small 0 may be.
Now when 0 is very small cos 0 is very nearly unity,
and the smaller 0 becomes, the more nearly does cos
become unity, and hence the more nearly does 1 -become unity.
become unity.