Plane trigonometry, by S.L. Loney.

ANGLES OF ANY SIZE AND SIGN.

69

The two triangles MOP and MJfP'O are therefore equal
in all respects.
Hence OM and M'P' are numerically equal, as also
HMP and OM' are numerically equal.
In each figure OM and M'P' have the same sign,
whilst MP and OM' have the opposite, so that
M'P'= + OM, and OM' =- JP.
We therefore have
M'P' OH
sin (90~ + O) = sin A OP' = P = cos,
OP1 OP
OH' - MP
cos (90~ 0) = cos AOP' = O = p = - sin 0,
M'P' OM
tan (90~ + 0) = tan A OP' = - = IP = - cot 0,
OM' - MP
cot (90~ + 0) =cot A OP = O' - tan 0,
OP' OP
see (90~ + 0) = sec A OP' = - A P =-P c-osec O,
OP' OP
and cosec (90~ + 0) = cosec A OP' = -- = O = sec0.
Exs. sin 150 = sin (90~ + 60~) = cos 60~ =,
cos 135 = cos (90~ + 45~) = - sin 45 = - 1
v./2
and tan 120~ = tan (90~ + 30~) = -cot 30 = - /3.
71. Supplementary Angles.
Two angles are said to be supplementary when their
sum is equal to two right angles, i.e. the supplement of
any angle 0 is 180~ -.