58 



PRACTICAL MATHEMATICS 



(c) cosh 2 a + sinh 2 a = ^ {(e a + e~ a ) z + (e a - e~ a ) 2 } 



Then 

 or 



(d) 



Putting 



= cosh 2a 



cosh 2a = 2 cosh 2 a 1 

 cosh 2a = 1 + 2 sinh 2 a 



cosh 2 a = - (cosh 2a + 1) 



sinh 2 a = - (cosh 2a 1) 



6= 2a 



Then cosh - = 



6 



u 



smh- = 



37. Tfo Series for sinh a and cosft a. 



cosh 0+1 



cosh 61 



Subtracting 



and 

 Adding 



and 



= 2 a 



+ . . . 



cosh a = - 







= 1 + . + 



38. It is important that we should be able to find the angle 

 when we are given one or other of its hyperbolic functions. 

 (a) If sinh <x = x 



Then \ (e" - e~ a ) = x 



e za - I = 2x 



_ 20; 



and a = log e { + 



