Synopsis :

1. i) sinhx = 2

ee xx

ii) coshx = 2

ee xx +

iii) tanhx = xxxx

eeee

+

iv) cothx = xxxx

eeee

+

v) sechx = xx ee2

+

vi) cosechx = xx ee2

are called hyperbolic

functions.

Note : sin(ix) = isinhx, cos(ix) = coshx, tan(ix) = itanhx.

2. cosh2x sinh2x = 1

3. 1 tanh2x = sech2x

4. coth2x 1 = cosech2x

5. i) sinh ( + ) = sinh cosh + cosh sinh ii) sinh ( ) = sinh cosh cosh sinh iii) cosh ( + ) = cosh cosh + sinh sinh iv) cosh ( ) = cosh cosh sinh sinh v) tanh ( + ) = +

+tanhtanh1tanhtanh

vi) tanh ( ) =

tanhtanh1tanhtanh

6. i) sinh 2x = 2sinhx coshx

ii) cosh 2x = cosh2x + sinh2x

iii) cosh 2x = 2 cosh2 x 1 or cosh2x = 2

x2cosh1+

iv) cosh 2x = 1 + sinh2x or 2sinh2x = 2

1x2cosh

v) tanh 2x = xtanh1

xtanh22+

HYPERBOLIC FUNCTIONS

Hyperbolic Functions

2

vi) sinh 3x = 3sinhx + 4sinh3x

vii) cosh 3x = 4cosh3x 3coshx

viii) tanh 3x = xtanh31

xtanhxtanh32

3

++

7. Values of inverse hyperbolic functions as logarithms functions :

i) )1xx(logxSinh 2e1 ++=

ii) 1 x),1xxlog(xCosh 21 +=

iii) 1) (-1, x,x1x1log

21xTanh 1

+=

iv) 1|x| ,1x1xlog

21xCoth 1 >

+=

v) 1x0 ,x

x11logxSech2

1

++= or

0 x,x

x11log2