3.38 



i. 



2. 



TRIGONOMETRY 



sinh 2x = 2 sinh x cosh x, 



2 tanh x 

 ~ i - tanh 2 x' 



cosh 2X = cosh 2 x + sinh 2 x = 2 cosh 2 x - i, 

 = 1 + 2 sinh 2 jc, 

 i + tanh 2 x 



75 



tanh 



i tanh 2 x 

 2 tanh x 



i + tanh 2 x 



sinh 30; = 3 sinh x + 4 sinh 3 x. 

 cosh 3# = 4 cosh 3 # 3 cosh #. 



3 tanh x + tanh 3 x 



tanh 



. 

 1 + 3 tanh 2 x 



3.40 Inverse Hyperbolic Functions. 



The hyperbolic functions being periodic, the inverse functions are multiple 

 valued (3.311). In the following formulas the periodic constants are omitted, 

 the principal values only being given. 



6. 



sinh" 1 x = log (x + Vx 2 + i) = cosh' 1 Vx 2 + i. 

 cosh" 1 x = log (x + Vx 2 - i) = sinh" 1 Vx 2 - i. 



tanh" 1 x = log V/ - 



* i x 



coth- 1 * = lo 



= tann" 1 - 

 x 



sech- 1 * = log(j + y 1 - ij = cosh- 1 ^- 

 csch" 1 x = log ( - + V -r + i ) = sinn" 1 



3 \X V ^ 2 / iJf 



3.41 



i. 



2. 



sinh" 1 x sinh" 1 y = 

 cosh" 1 x cosh" 1 3; = cosh" 1 (xy V(x 2 - i)(/ - i)), 



a; v 



tanh" 1 



tanh" 1 



tanh 1 



