Plane trigonometry, by S.L. Loney.

HYPERBOLIC FUNCTIONS.

369

EXAMPLES. LIV.
exi + e-i ezi - e-xi
Assuming that cos x= and sinx= 2i prove that, for
2 22
all values of x, real or complex,
1. cos2 x + sin2 x =. 2. cos(-x) =cosx.
3. sin(- x)=sinx. 4. cos 2x=cos2 x -sin2 x =1-2 sin2x.
5, sin3x=3 sinx-4sin3. 6.cosx-cosy2 sin sin
2 2
x+y x7. sin x- sin y= 2 cos sin.
2 2
Prove that
8. {sin(a+0) -ei sin =sinna e -n.
9, sin (a + nO) - eai sin nO = e-nei sin a.
10. {sin (a- )+e-ai sin }= sinn-l a { sin (a — n) + eai sin nO.
312. In the formulae of Art. 308 if x be a pure
imaginary quantity and equal to yi, we have, since
i2=- 1,
eyi.-ie-yi e-y + e ey + ecos yI = =,
and
eyi.i - e-yi i e — ey e-y - ey
sin yi =2i
2smyz=- 2 = 2(- 1). ey - e-y
2
313. Hyperbolic Functions. Def. The quantity
eY- e-^
2
whether y be real or complex, is called the hyperbolic
sine of y and is written sinh y.
L. T. 24