Plane trigonometry, by S.L. Loney.

ANGLES OF ANY SIZE AND SIGN.

65

Let the revolving line, starting from OA, revolve
through any angle 0 and stop in the position OP.
Draw PM perpendicular to OA (or OA produced) and
produce it to P', so that the lengths of PM and MP' are
equal.
In the geometrical triangles MOP and MOP' we have
the two sides OM and MP equal to the two OM and
MP', and the included angles OMP and OMP' are right
angles.
Hence (Euc. I. 4), the magnitudes of the angles MOP
and MOP' are the same and OP is equal to OP'.
In each of the four figures, the magnitudes of the
angle AOP (measured counter-clockwise) and of the angle
AOP' (measured clockwise) are the same.
Hence the angle A OP' (measured clockwise) is
denoted by -0.
Also MP and MP' are equal in magnitude but are
opposite in sign. (Art. 49.) We have therefore
MP' - MP
sin ( — o) = o - p=-sin
'OM OM
cos (- ) = == 0 =cos,
MP' -MP
tan(-))=-O-3 OM =-tan,, AOM OM
co(- ) = - cot 0,
OP' OM1.T
cosec ( - 0) -= = P P cosec 0
OP' OP
and sec ( - 0) = = = sec 0.
LM O. S

L. T.

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