Plane trigonometry, by S.L. Loney.

372

TRIGONOMETRY.

Again, for all values of 0, we have
3 tan 0- tan3 0
tan 36 =
1 - 3 tan2 0
Put then 0 = xi, and we have
tan (3xi) 3 tan (xi) - tan3 (xi)
tan (3X)=1 - 3 tan2 (xi)
Hence the substitutions of Art. 314 give
3i tanh x - i3 tanh3 x
tanh (3x) = 1 - 3i2 tanh2 x
3i tanh x + i tanh3 x
1 + 3 tanh2 x
3 tanh x + tanh3 x
so that tanh (3x) = 1 +3 tanh2 x
I + 13 tanh2 X
As before, this may be easily proved from the definition
of tanh x.
316. In general it follows from (1) of Art. 314 that
any general formula which is true for cosines of angles is
also true if instead of cos we read cosh.
From (2) of the same article, since
sin2 (yi) = - sinh2 y,
it follows that any general formula involving the cosine
and square of the sine of an angle is true if for cos we
read cosh and for sin2 we read - sinh2.
Similarly from (3) we may turn a formula involving
tan2 into another by writing for tan2 the quantity - tanh2.
In this manner formulae and series involving the
hyperbolic functions may be obtained from Arts. 241,
242, 274, 275, 277, 289, 291, and 293-298.