Basic Hyperbolic Simplification

**ugkwan** · September 30th 2010, 11:11 PM

Hi,

question:
$tanh(-2)=?$
I tried simplifying it and got
$\frac{sinh(-2)}{cosh(-2)}=\frac{\frac{e^2}{1}-\frac{1}{e^2}}{\frac{e^2}{1}+\frac{1}{e^2}}=1+e^4-\frac{1}{e^4}-1=e^4-\frac{1}{e^4}$
but this calculates into 54.579 when the answer is -.964
I could use a calculator and figure out the answer from the point
$\frac{e^{-2} - e^2}{e^{-2} + e^2}$
Im trying to figure out where my algebra mistake is b/c my professor forbids calculators on our test which is tomorrow. Help?

**Prove It** · September 30th 2010, 11:37 PM

$\displaystyle{\tanh{x} = \frac{e^{2x} - 1}{e^{2x} + 1}}$ .

So what does $\tanh{(-2)}$ equal?

**HallsofIvy** · October 1st 2010, 02:14 AM

Of course, the definition of "tanh" is $tanh(x)= \frac{sinh(x)}{cosh(x)}= \frac{\frac{e^x- e^{-1}}{2}}{\frac{e^x+ e^{-x}}{2}}= \frac{e^x- e^{-x}}{e^x+ e^{-x}}$ as you have.

What Prove It did was multiply both numerator and denominator by $e^x$ to get $tanh(x)= \frac{e^{2x}- 1}{e^{2x}+1}$

You are (almost) correct to $\frac{\frac{e^2}{1}-\frac{1}{e^2}}{\frac{e^2}{1}+\frac{1}{e^2}}$ (each "2" should be "-2") but I cannot imagine how you then wrote that fraction as four added and subtracted terms If you multiply numerator and denominator of
$\dfrac{\frac{e^{-2}}{1}-\frac{1}{e^{-2}}}{\frac{e^{-2}}{1}+\frac{1}{e^{-2}}}$
by $e^2$ you get
$\dfrac{e^{-4}- 1}{e^{-4}+ 1}$ .

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