簡單SINH function

Hyperbolic sine is defined as:

e^x – e^(-x) e^2x - 1
sinh x = --------------- = ---------------
2 2e^x

sinh^-1 (2) means

e^2x - 1
sinh x = -------------- = 2
2e^x

You have to find the value of x such that sinh x = 2

e^2x - 1
--------------- = 2
2e^x

e^2x - 1 = 2(2e^x)
(e^x)^2 - 1 = 4e^x
(e^x)^2 - 4e^x – 1 = 0
Let y = e^x
y^2 - 4y – 1 = 0
Using quadratic formula
y = {-(-4)+-[(-4)^2 – 4(1)(-1)]^0.5}/2(1)
y = 4+-[20]^0.5}/2
y =4.2361 or -0.2361 (reject negative number)
e^x = 4.2361
Taking natural log on both sides
Ln(e^x) = ln 4.2361
x ln e = ln 4.2361
x (1) = 1.4436
x = 1.4436

In some scientific calculators, they have built-in function of sinh. You can find the solution by pressing the key. It is very fast.
In the old days, you can also obtain the answer by looking up the four-figure tables under the section sinh x, but it gives the answer up to 2 decimal places.
For example x = 1.44 (page 42, table sinh x,
“four-figure tables” – by C. Godfrey and A.W. Siddons, Cambridge University Press.

To回答者： Ch.alex
You stated that sinh^-1(2) =2e^2/(e^4-1)　
= 2(2.7182828)^2/(2.71828)^4 – 1) Note: taking e = 2.7182828
= 2(7.389)/(54.59815 -1)
= 14.7781/53.5981
= 0.27577

It does not give the answer of 1.4436 as you stated.
You got the correct answer. No offence:Did you use the built-in function of the calculator?
sinh^-1(2) is an inverse function of sinh x, not reciprocal of sinh x.
You statement sinh^-1(2) =2e^2/(e^4-1) is wrong because you assume that sinh^-1(2) = 1/sinh 2

sinh^-1(x)=(2e^x)/(e^2x - 1)

sinh^-1(2)=2e^2/(e^4-1)　(用e表達就係最終答案)

　　　　 =1.4436 (corr. to 5 sig-fig)