Plane trigonometry, by S.L. Loney.

TRIGONOMETRY.

97. The formulae (1), (2), (3), and (4) of Art. 94 are
also very important. They should be remembered in the
form
2sin A cos B = sin (A + B) + sin (A - B)...(1),
2 cos A sin B = sin (A + B) - sin (A - B)...(2),
2 cos A cos B = cos (A + B) + cos (A - B)...(3),
2 sin A sin B =cos (A - B) - cos (A + B)...(4).
They may be looked upon as the converse of the
formulae I-IV. of Art. 94.
Ex. 1. 2 sin 30 cos 0= sin 40 + sin 20.
Ex. 2. 2 sin 50 sin 30= cos 20 - cos 80.
Ex. 3. 2 cos 110 cos 20 = cos 130 + cos 90.
Ex. 4. Simplify
sin 80 cos 0 - sin 60 cos 30
cos 20 cos 0 - sin 30 sin 40
By the above formulae the expression
1 [sin 90 + sin 70] - [sin 90 + sin 30]
21 1
- [cos 30 + cos 0] - - [cos 0 - cos 70]
2 2
sin 70- sin 30
cos 30 + cos 70
2 cos 50 sin 20
2= _os 580 s 20 by the formulae of Art. 94,
2 cos 50 cos 20 '
= tan 20.
[The student should carefully notice the artifice of first employing
the formulae of this article and then, to obtain a further simplification,
employing the converse formulae of Art. 94. This artifice is often
successful in simplifications.]
EXAMPLES. XV.
Express as a sum or difference the following
1. 2 sin 50 sin 70. 2. 2 cos 70 sin 50.
3. 2cos 110 cos 30. 4. 2 sin 54~ sin 66~.