Plane trigonometry, by S.L. Loney.

TRIGONOMETRICAL RATIOS.

21

OP'M' are the same as those derived from the triangle
OPM.
In the two triangles the angle at 0 is common, and
the angles at M and M' are both right angles and therefore equal.
Hence the two triangles are equiangular and therefore,
MP M'P
by Euc. VI. 4, we have Op - O' ' i.e. the sine of the angle
A OP is the same whatever point we take on the revolving
line.
Since, by the same proposition, we have
OM OM' MP M'P'
OP= ~ and OM - O- -
it follows that the cosine and tangent are the same
whatever point be taken on the revolving line. Similarly
for the other ratios.
If OA be considered as the revolving line and in it be taken any
point P" and P"M" be drawn perpendicular to OP, the functions as
derived from the triangle OP"M" will have the same values as before.
For, since in the two triangles OPM and OP"M", the two angles
P"OM" and OM"P" are respectively equal to POM and OMP, these
two triangles are equiangular and therefore similar, and we have
M"P" MP OM" OM
Or" =Op' and
OP " = OP ' OP" - OP'
27. Fundamental relations between the trigonometrical
ratios of an angle.
We shall find that if one of the trigonometrical ratios
of an angle be known, the numerical magnitude of each of
the others is known also.
Let the angle AOP (Fig., Art. 23) be denoted by 0.
In the triangle AOP we have, by Euc. I. 47,
MP + OM = OP2...................(1).