Plane trigonometry, by S.L. Loney.

28 TRIGONOMETRY.
OM x
Hence cot 0 = P = = x,
MP 1 1
sin 0 - _ ____
OP V1 +2 1 + Cot20'
OM x cot 0
COS 0 =
OP ~/1 -g + ~2 V1+ cot20'
MP 1 1
tan 0 = - -=
OM x cot O'
OP /_ Vi+x2 /1 +cot20
sec 0 = cot
OM x cot 0
OP \/1.-v x
and cosec 0 = - 1 = /1 = V + cot20.
Jf^ 1
The last five equations give what is required.
It will be noticed that, in each case, the denominator
of the fraction which defines the trigonometrical ratio was
MP
taken equal to unity. For example, the sine is -p, and
hence in Ex. 1 the denominator OP is taken equal to
unity.
Om
The cotangent is MRp, and hence in Ex. 2 the side MP
is taken equal to unity.
Similarly suppose we had to express the other ratios
in terms of the cosine, we should, since the cosine is equal
Om
to p, put OP equal to unity and OM equal to c. The
working would then be similar to that of Exs. 1 and 2.
In the following examples the sides have numerical
values.