Plane trigonometry, by S.L. Loney.

TRIGONOMETRICAL RATIOS.

27

The simplest method of procedure is best shewn by
examples.
Ex. 1. To express all the trigo~no-
metrical ratios in terms of the sine.
Let A OP be any angle 0. e \ I
Let the length OP be unity and let
the corresponding length of MP be s.
By Euc. I. 47, OM = /OP _- Mp2 = Vi - s2.

Hence

MP s
sin0= =-=s,
OMcos 0 =
cos 0 = O /1 - s2 = -V1 - sin2 0,
OP

MP s sin 0
tan 0 =.
tan0S= 0-M/I- 82 = s -sin20'
OM Vl- s 2/1- sin2 0
cot /= -MP= e - Sin '
cosec 0= P = =
MP s sin '
OP 1 1
cosec0= = ssin0
OP 1 1
sece 0= O-
OMV1 ' -2s /1-sin2''

and

The last five equations give what is required.
Ex. 2. To express all the trigonometrical relations in
terms of the cotangent.
Taking the usual figure let the
length MP be unity, and let the corre- P
sponding value of OM be x.
By Euc. I. 47, o X,- A
OP = 0/OMP2 + MP2= /1 +.