Plane trigonometry, by S.L. Loney.

132

TRIGONOMETRY.

Alotn( + B+ ) = tan (A 4+ B) + tan C
I - tan (A + B) tan C
tan A + tan B
-+ tan C
- tan A tan B
=,.......
tan A + tanB
1 - tan C
1 - tanA tanB
tan A + tan B + tan C - tan A tan B tan C
- tan B tan C - tan C tan A - tan A tan B'
125. The last formula of the 'previous article is a
particular case of a very general theorem which gives the
tangent of the sum of any number of angles in terms of
the tangents of the angles themselves. The theorem is
tan (A1 + A2 + A3 +... + A,,)
S1 -S3 + - 7...(1),
3 +;4 5- S +.*................(
1 -S2 + S4- S6 +...
where
s, = tan A, + tan A2 +... + tan An
= the sum of the tangents of the separate angles,
2 = tan A, tan A2 + tan A, tan As +...
= the sum of the tangents taken two at a time,
s3 = tan Al tan A, tan A3 + tan A2 tan A3 tan A4 +...
= the sum of the tangents taken three at a time, and so
on.
Assume the relation (1) to hold for n angles and add
on another angle An+i.
Then tan (Al + A2 +... + An+l)
= tan [(A, + A, +... + An) + A,+l]
tan (A + A, +... + An) + tan A,+
1 - tan (AI + A2 +... + A,,). tan An,+