Plane trigonometry, by S.L. Loney.

32

TRIGONOMETRY.

Values of the trigonometrical ratios in
some useful cases.
33. Angle of 45~.
Let the angle A OP traced out p
be 45~.
Then, since the three angles of
a triangle are together equal to
two right angles,45
0 M A
z OPM = 180~- z POM- L PMO
= 180~ - 45~- 90~= 45~ =Z POM... OM = MP = a (say),
and OP = VOM2 + MP2= /2. a.
MP a 1..sin 4O = - 2. a \/2'
OM a 1
cos 45= ~OP= /2. a ~/2'
and tan 45 = 1.
34. Angle of 300. p
Let the angle A OP traced
out be 30~.
Produce PM to P' making
MP' equal to PM.
The two triangles OMP and
OMP' have their sides OM and "
MP' equal to OM and MP and p
also the contained angles equal.
Therefore OP' = OP, and z OP'P = z OPP'= 60~, so
that the triangle P'OP is equilateral.