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Identities (formulas) for hyperbolic functions

Osborn's rule : In any formula connecting the circular function of general angles, replace each circular function by the corresponding hyperbolic function and change the sign of every product (or square) of two sines directly or indirectly involving in it and conversely.

Using this rule

e.g 1)

2)

Formulas

a)    1) cos h2x - sinh2x = 1      2) sec h2x = 1- tan h2x
       3) cosec h2x = 1 + cot h2x

b)    1)   sin h (a ± b ) = sin ha cos hb ± cos ha sin hb

       2)   cos h (a ± b ) = cos ha cos hb ± sin ha sin hb

       3)   tan h ( a ± b ) =

tan ha ± tan hb

1 ± tan ha tan hb

   d)   1)    sin h2x   =   2 sin hx cos hx

         2)  cos h2x    =   cos h2x + sin h2x
                              =   2 cos h2x-1
                              =   1 + 2 sin h2x

        3)    tan h2x   =   

   e)   1)   cos h2x = 2) sin h2x =

   f)   1)   sin h3x = 3 sin hx + 4 sin h3x      2) cos h3x = 3 cos hx - 4 cos h3x

Inverse Hyperbolic Functions

If x = cos hy then we write y = cos h-1x

If x be real, we have

      +ve value of the R.H.S. is always taken.

   Similarly it can be shown, if x is real

Differentiation and integration

(1) [ sin hx] = cos hx            \ (1) ò cos hx dx = sin hx

(2) [cos hx] = sin hx             \ (2) ò sin hx dx = cos hx

(3) [ tan hx ] = sec h2x          \ (3) ò sec h2x dx = tan hx

(4)    
\ (4) ò

(5)
(5) ò

(6) =
\ (6) ò =

(7) =
\ (7) ò =

(8) = \ (8) ò =

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Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

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