Plane trigonometry, by S.L. Loney.

144

TRIGONOMETRY.

As 0 increases from 0 to 45~, sin (0 + 45~) increases
from sin 45~ to sin 90~, and hence the expression increases
from 1 to /2.
As 0 increases from 45~ to 135~, 0+ 45~ increases from
90~ to 180~, and hence the expression is positive and
decreases from /2 to 0.
As 0 increases from 135~ to 225~, the expression
changes from 1/2 sin 180~ to 1/2 sin 270~, i.e. it is negative
and decreases from 0 to - /2.
As 0 increases from 225~ to 315~, the expression
changes from /2 sin 270~ to /2 sin 360~, i.e. it is negative
and increases from - \/2 to 0.
As 0 increases from 315~ to 360~, the expression
changes from 1/2 sin 360~ to /2 sin 405~, i.e. it is positive
and increases from 0 to 1.
133. Ex. To trace the changes in the sign and
magnitude of a cos 0 + b sin 0, and to find the greatest
value of the expression.
We have
a b
a cos 0 + b sin 0 = a2 + b22 a 2 cos 0 + b sin 0
Let a be the smallest positive angle such that
a b
cos a,and sin a =
Va2 + b2 2/a2+b2
The expression therefore
= /a2 + b2 [cos 0 cos a + sin 0 sin a] = Va2 + b2 cos (0 - a).
As 0 changes from a to 360 ~+a, the angle 0-a
changes from 0 to 360~, and hence the changes in the
sign and magnitude of the expression are easily obtained.