3 Related hyperbolic functions

Given the trigonometric functions cos x , sin x related functions can be defined; tan x , sec x , c o s e c x through the relations:

tan x sin x cos x sec x 1 cos x c o s e c x 1 sin x cot x cos x sin x

In an analogous way, given cosh x and sinh x we can introduce hyperbolic functions tanh x , s e c h x , c o s e c h x and coth x . These functions are defined in the following Key Point:

Key Point 6

Further Hyperbolic Functions

tanh x sinh x cosh x sech x 1 cosh x c o s e c h x 1 sinh x coth x cosh x sinh x
Task!

Show that 1 tanh 2 x sech 2 x

Use the identity cosh 2 x sinh 2 x 1 :

Dividing both sides by cosh 2 x gives

1 sinh 2 x cosh 2 x 1 cosh 2 x implying (see Key Point 6) 1 tanh 2 x sech 2 x

Exercises
  1. Express
    1. 2 sinh x + 3 cosh x in terms of e x and e x .
    2. 2 sinh 4 x 7 cosh 4 x in terms of e 4 x and e 4 x .
  2.  Express
    1. 2 e x e x in terms of sinh x and cosh x .
    2. 7 e x ( e x e x ) in terms of sinh x and cosh x , and then in terms of coth x .
    3. 4 e 3 x 3 e 3 x in terms of sinh 3 x and cosh 3 x .

      3. Using only the cosh and sinh keys on your calculator (or e x key) find the values of

  3. tanh 0.35 ,
  4. c o s e c h 2 ,
  5. sech 0.6 .
    1. 5 2 e x 1 2 e x
    2.   5 2 e 4 x 9 2 e 4 x
    1. cosh x + 3 sinh x ,
    2. 7 ( cosh x + sinh x ) 2 sinh x , 7 2 ( coth x + 1 )
    3.   cosh 3 x 7 sinh 3 x
  1. 0.3364,
  2. 0.2757
  3. 0.8436