Plane trigonometry, by S.L. Loney.

80

TRIGONOMETRY.

Draw PM perpendicular to OA and pro- --—.
duce it to P', making PM equal to MP'.
When the revolving line is in the position
OP or OP', and in no other position, then, as
in Art. 78, the cosine of the angle traced out '
is equal to the given cosine.
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. it has described an angle 21zrT + a,
where n is zero or some positive or negative integer.
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. it has described an angle 2n7r-a.
All these angles are included in the expression
2n7r + a.......................1)
where n is some positive or negative number.
Cor. The expression (1) includes all angles having
the same secant as a.
84. Theorem. To find a general expression for all
angles which have the same tangent.
Let A OP be the snallest angle having the given

tangent, and let it be denoted by a.
Produce PO to P' making OP'
equal to OP and draw P'M' perpendicular to 01M'.
As in Art. 73 the angles AOP
and AOP' have the same tangent;
also the angle A OP' = r + a.
When the revolving line is in

P-..,
I l
II I -4~A
0,-" V