Plane trigonometry, by S.L. Loney.

116 TRIGONOMETRY.
i.e. (sin A + cos = 1 + sin A,
and (sin A -cos ) = 1 - sin A;
A A
so that sin -+ cos = + i 1 + sin A............(3),
A A
and sin - cos - =~/ - sin A.............(4).
By adding, and then subtracting, we have
2 sin = + /+sinA + /I -sin A...... (5),
and 2 cos A = + 1 + sin A!1 - sin A......(6).
A
The other ratios of A are then easily obtained.
114. In each of the formulae (5) and (6) there are
two ambiguous signs. In the following examples it is
shewn how to determine the ambiguity in any particular
case.
Ex. 1. Given that sin 300 is I, find the values of sin 15~ and cos 15~.
2i
Putting A= 30, we have from relations (3) and (4),
sin 150 + cos 150= A \/1+sin 30~= 4 v3
__ 1
sin 15~ - cos 15~= = /r1 - sin 30~-=: 2.
412
Now sin 15~ and cos 15~ are both positive and cos 15~ is greater than
sin 15~. Hence the expressions sin 15~ + cos 15~ and sin 15~ - cos 15~ are
respectively positive and negative.