Plane trigonometry, by S.L. Loney.

78

TRIGONOMETRY.

Similar statements are true if the cosine, tangent, or
any other trigonometrical function of the angle be given.
Hence, simply to give one of the trigonometrical
functions of an angle does not determine it without
ambiguity.
81. Suppose we know that the revolving line OP
coincides with the initial line OA. All we know is that
the revolving line has made 0, or 1, or 2, or 3,... complete
revolutions, either positive or negative.
But when the revolving line has made one complete
revolution the angle it has described is (Art. 17) equal to
27r radians.
Hence when the revolving line OP coincides with the
initial line OA, the angle that it has described is 0, or 1,
or 2, or 3... times 27r radians, in either the positive or
negative directions, i.e. either 0, or + 27r, or + 47r, or + 6wr...
radians.
This is expressed by saying that when the revolving
line coincides with the initial line the angle it has described is 2n7r, where n is some positive or negative
whole number.
82. Theorem. To find a general expression to include all angles which have the same sine.
Let AOP be the smallest positive angle having the
given sine and let it be denoted
by a. -- "-,
Draw PM perpendicular to OA p
and produce 3IMO to M' making /
1M0 equal to OM' and draw M'P' A' i' o 0:A
parallel and equal to MP.
As in Art. 72 the angle AOP' is equal to r - a.