If tanh(x) = 5/13, find the values of the other hyperbolic functions at x

To find the values of the other hyperbolic functions given that tanh(x) = 5/13, we can utilize the definitions of the hyperbolic functions in terms of ex.

The hyperbolic functions are defined as follows:

• sinh(x) = (ex - e-x) / 2
• cosh(x) = (ex + e-x) / 2
• tanh(x) = sinh(x) / cosh(x)

From the definition of the hyperbolic tangent, we know:

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

Given that tanh(x) = 5/13, we can let:

sinh(x) = 5k  

cosh(x) = 13k

for some positive constant k. Now we can find k using the identity sinh2(x) + cosh2(x) = 1:

(5k)2 + (13k)2 = 1

25k2 + 169k2 = 1

194k2 = 1

k2 = 1/194

k = 1/√194

Now substituting back to find sinh(x) and cosh(x):

sinh(x) = 5/√194  

cosh(x) = 13/√194

Finally, we can find the values of the other hyperbolic functions:

• tanh(x) = 5/13
• sinh(x) = 5/√194
• cosh(x) = 13/√194
• sech(x) = 1/cosh(x) = √194/13
• csch(x) = 1/sinh(x) = √194/5
• coth(x) = cosh(x)/sinh(x) = 13/5

So, the final results are:

• sinh(x) = 5/√194
• cosh(x) = 13/√194
• sech(x) = √194/13
• csch(x) = √194/5
• coth(x) = 13/5