How do I solve sinh(x)=1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that `(e^x - e^(-x))/2`  gives the hyperbolic sine, hence, you need to solve the equation `(e^x - e^(-x))/2 = 1`  such that:

`(e^x - e^(-x))/2 = 1 =gt (e^x - e^(-x)) = 2`

You need to write `e^(-x)`  making use of negative power property such that:

`e^(-x) = 1/e^x`

`e^x - 1/e^x - 2 = 0`

You need to bring the terms to a common denominator such that:

`e^(2x) - 2e^x - 1 = 0`

You should come up with the substitution`e^x = t`  such that:

`t^2 - 2t - 1 = 0`

`t_(1,2) = (2+-sqrt(4+4))/2`

`t_1 = (2+2sqrt2)/2 =gt t_1 = 1+sqrt2`

`t_2 = 1-sqrt2`

You need to solve for x the equations `e^x = t_1`  and `e^x = t^2`  such that:

`e^x = 1+sqrt2 =gt x = ln(1+sqrt2)`

`e^x = 1-sqrt2 lt 0`  impossible because `e^x`  needs to be strictly positive.

Hence, evaluating the solution to equation sinh = 1 yields `x = ln(1+sqrt2).`

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