If cosh(x) = 5/3 and x > 0, find the values of the other hyperbolic functions at x.

Question

If cosh(x) = 5/3 and x > 0, find the values of the other hyperbolic functions at x.

 

I tried solving for x, but that didn't work

 

p262, #21

ANSWER:

sinh(x) = 4/3

sech(x) = 3/5

csch(x) = 3/4

tanh(x) = 4/5

coth(x) = 5/4

Answer 1

The hyperbolic functions have very similar properties to their trigonometric counterparts.  They satisfy the following relations:

 

cosh(x)² - sinh(x)² = 1

sech(x) = 1/cosh(x)

csch(x)  = 1/sinh(x)

tanh(x) = sinh(x)/cosh(x)

coth(x) = 1/tanh(x)

 

Using these relations we have and the fact that cosh(x) = 5/3, we have:

 

1. (5/3)² - sinh(x)² = 1

sinh(x)² = (5/3)² -1 = 25/9 - 1 = 16 / 9

then sinh(x) = sqrt(16/9) = 4/3

 

2. sech(x) = 1/cosh(x) = 1/(5/3) = 3/5

 

3. csch(x) = 1/sinh(x) = 1/(4/3) = 3/4

 

4. tanh(x) = sinh(x)/cosh(x) = (4/3) /(5/3) = 4/5

 

5. coth(x) = 1/tanh(x) = 1/(4/5) = 5/4