Plane trigonometry, by S.L. Loney.

108

TRIGONOMETRY.

tan A + tan 2A
Also tan 3A = tan (A + 2A) tan A +tan 2A
2v 1 - tan A tan 24
2 tan A
tanA +1 - tan2 A tan (1 - tan2 A) + 2 tan A
1-tanA 2 tanA (1 - tan2 A) - 2 tan2 A
1- an *1 - -tan2 A1-tan A
3 tan A - tan3 A
Hence tan 3A = 1-3tan2A
1 - 3 tan2 A
108. By a process similar to that of the last article,
the trigonometrical ratios of any higher multiples of 0
may be expressed in terms of those of 0. The method is
however long and tedious. In a later chapter better
methods will be pointed out.
As an example let us express cos 50 in terms of cos 0.
We have
cos 50 = cos (30 + 20)
= cos 30 cos 20- sin 30 sin 20
= (4 cos3 0 - 3 cos 0) (2 cos2 0 - 1)
- (3 sin 0 - 4 sin3 0). 2 sin 0 cos 0
= (8 cos5 0 - 10 cos + 3 cos 0)
-2 cos 0. sin2 0(3 - 4 sin2 0)
= (8 cos5 0 - 10 cos3 0 + 3 cos 0)
- 2 cos 0 (1 - cos2 0) (4 cos2 0-1)
= (8 cos5 0- 10 cos3 0 + 3 cos 0)
- 2 cos 0 (5 cos2 0- 4 cos4 0- 1)
= 16 cos5 0 -20 cos3 0+ 5 cos 0.