What Is the Solution to sinh(x) = 1?

[kd001]

Homework Statement

sinh(x) = 1

What is the value of 'x'?

Homework Equations

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

The Attempt at a Solution

e^x - e^-x = 2

Then what do I do?

Thanks

 

[emanuel_hr]

Multiply the whole eq by e^{x} then solve the resulting quadtatic eqn in e^{x} and afterwards keep the positive solution and take its logarithm to obtain x.

 

[arkajad]

Multiply the whole eq by e^{x} then solve the resulting quadtatic eqn in e^{x}

That is put, say, u=e^x and solve for u. Then find x from e^x=u.

 

[kd001]

Thanks. That worked.

 

[HallsofIvy]

Homework Statement

sinh(x) = 1

What is the value of 'x'?

Homework Equations

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

The Attempt at a Solution

e^x - e^-x = 2

Then what do I do?

Thanks

Multiply on both sides by e^x to get (e^x)^2- 1= 2e^x and then subtract 2e^x- 1 from both sides: (e^x)^2- 2e^x= 1. Think of that as a quadratic equation in e^x and complete the square: (e^x)^2- 2e^x+ 1= (e^x- 1)^2= 2.

Take the square root of both sides, e^x- 1= \pm\sqrt{2} and add 1 to both sides, e^x= 1\pm\sqrt{2}. 1- \sqrt{2}< 0 so you get the single solution x= ln(1+ \sqrt{2}).

Too slow! Too slow!