Solve for x if cosh(x) = 0.5 / tanh(x)?

Question

I have to do this equation step by step...I have a few answers but I don't understand what is happening in each step...can someone solve this equation and explain step by step what is happening....so far I have 2 different answers...

1) cosh(x) = 0...but it can't equal 0 (why?..i don't understand)

2) people sustituted the cosh(x) for a 1, then solved the equation...WHY did they substitute the cosh(x) for a 1...what's the rule there...

Thank you...

Answer 1

cosh(x) * tanh(x) = sinh(x) = 0.5
x = asinh(0.5) = 0.4812

If you want an exact answer, use the definition of sinh(x)
sinh(x) = [e^x - e^(-x)]/2 = 0.5
e^x - 1/e^x = 1
e^2x - e^x - 1 = 0
e^x = [1+sqrt(5)]/2
x = ln{ [1+sqrt(5)]/2 }

*EDIT*
I don't know who solved your equations so I can't comment too much on that.
cosh(x) cannot be zero because
cosh(x) = [e^x + e^-x]/2
Both terms in the [ ] are positive, so cosh can never be negative or zero. In fact the minimum value of cosh(x) is 1.

Answer 2

1. cosh(x) = 0.5/something. A fraction equals zero only when the numerator is equal to zero. Since the numerator is 0.5, it is never zero.

2. I have no idea why the others did that. It isn't a valid thing to do unless it was given in the problem that cosh(x) = 1.

tanh(x) = sinh(x)/cosh(x)
Substitute this in the equation and it will be easy to solve.

Answer 3

tanh= sinh/cosh
so
cosh * tan h = sinh=0.5
without calculator
e^x-e^-x =1
Call e^x=z
z-1/z=1 z^2-z-1 =0
z= ((1+sqrt(5))/2 (remember z must be >0)and x=ln[(1+sqrt5)/2]