Plane trigonometry, by S.L. Loney.

ANGLES OF ANY SIZE AND SIGN.

71

Take OP' equal to OP and draw P'M' and PM
perpendicular to AOA'.
The angles MOP and M'OP' are equal and hence the
triangles MOP and M'OP' are equal in all respects.
Hence OM and OM' are equal in magnitude and so
also are MP and M'P'.
In each figure OM and OM' are drawn in opposite
directions, whilst MP and M'P' are drawn in the same
direction, so that
OM' =- OM, and M'P'= + MP.
Hence we have
M'P' MP
siin(18O~-)=sinAOP'=- O-=sin 0,
OM' -0OM
cos ( 18 - )=cos A OP'= O,= OP =-cos,
M'P' MP
tan (180~ -0)= tan A P'= 0,' _ OM-P - tan 0,
cot (180~ - )=cot AOP'=- M', = 'M - cot,
OP' OP
sec (80~ - 0) = sec A OP'= -O = OM - - sec 8,
OP' OP
and cosec (180~ - ) = cosec A OP' = i = = cosec.
MP3
Exs. sin 120~= sin (180~ - 60~) =sin 60~= 2,
cos 135~= cos (180~ - 45~) = - cos 45~ =
a/2'
and tan 1500 = tan (180~ - 30~) = - tan 30~ = -
VS.