Plane trigonometry, by S.L. Loney.

HYPERBOLIC FUNCTIONS.

371

315. Corresponding to most general trigonometrical
formulae involving the ratios of angles there are formulae
involving the hyperbolic ratios.
For example, we have, for all values of the angle x,
cos2 x + sin2 x= 1,
so that cos2 (yi) + sin2 (yi) = 1,
and hence, by the last article,
cosh2 y - sinh2 y = 1.
[This may be deduced independently from the definition of the hyperbolic functions. For
cosh2 y - sinh2 y (ey + e-y 2 - (e -e-)2
e2y 2 e-2y e2y - 2+ e-2y
4 - 4 =
Again, for all values of u and v we have
sin (u + v) = sin u cos v + cos it sin v.
Put u = xi and v = yi,
so that
sin [(x + y) i] = sin (xi) cos (yi) + cos (xi) sin (yi).
The expressions of the last article then give
i sinh (x + y) = i sinh x cosh y + cosh x x i sinh y,.. sinh (x + y) = sinh x cosh y + cosh x sinh y.
[Directly from the definition of the hyperbolic ratios we have
sinh x cosh y + cosh x sinh y
ex eX e-x e-Y e +-X ey - eY-y 2ex+Y - 2e-(X+Y)
2 2 2 2 4
on multiplication, = sinh (x + y).]
24-2