Plane trigonometry, by S.L. Loney.

ANGLES HAVING THE SAME SINE.

79

When the revolving line is in either of the positions
OP or OP', and in no other position, the sine of the angle
traced out is equal to the given sine.
When the revolving line is in the position OP it has
made a whole number of complete revolutions and then
described an angle a, i.e. by the last article it has described
an angle equal to
2r7r + a...........................(1)
where r is zero or some positive or negative integer.
When the revolving line is in the position OP' it has,
similarly, described an angle 2r7r + A OP', i.e. an angle
2r7r + 7r - a,
i.e. (2r+ 1)r - a...................(2)
where r is zero or some positive or negative integer.
All these angles will be found to be included in the
expression
n7r +( - l) a.......................(3),
where n is zero or a positive or negative integer.
For, when n = 2r, since (- 1)2 = + 1, the expression (3)
gives 2r~r + a, which is the same as the expression (1).
Also, when n = 2r + 1, since (- 1)2r+ =- 1, the expression (3) gives (2r-+ 1)r-a, which is the same as the
expression (2).
Cor. Since all angles which have the same sine have
also the same cosecant, the expression (3) includes all
angles which have the same cosecant as a.
83. Theorem. To find a general expression to include all angles which have the same cosine.
Let AOP be the smallest angle having the given
cosine and let it be denoted by a.