Plane trigonometry, by S.L. Loney.

ANGLES HAVING THE SAME TANGENT.

81

the position OP, it has described a whole number of
complete revolutions and then turned through an angle
a, i.e. it has described an angle
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 + (7r + a),
i.e. (2r + 1) r + a.....................(2).
All these angles are included in the expression
n7r+ a.............................(3).
For, when n is even, (= 2r say), the expression (3)
gives the same angles as the expression (1).
Also, when n is odd, (= 2r + 1 say), it gives the same
angles as the expression (2).
Cor. The expression (3) includes all angles which
have the same cotangent as a.
85. Ex. 1. Write down the general expression for all angles,
(1) whose sine is equal to,/3
1
(2) whose cosine is equal to -
1
and (3) whose tangent is equal to -.
(1) The smallest angle, whose sine is, is 60~, i.e. r.
Hence, by Art. 82, the general expression for all the angles which
have this sine is
nr-+(- l) +
(2) The smallest positive angle, whose cosine is, is 120~ i.e. 2
L. T63.

L. T.

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