How to remember hyperbolic functions

Question

I forget them all the time in solving PDE. Can someone provide a way to remember them: $$ \cosh\left(x\right)=\dfrac{e^x+e^{-x}}{2} \qquad \text{ and } \qquad \sinh\left(x\right)=\dfrac{e^x-e^{-x}}{2} $$

Answer 1

Is your problem remembering the formula itself, or just which of the two has the "+" and which has the "-" ?

If the latter, one would be

cosh == positive

sinh == negative

Answer 2

The definition does not help much at beginning, I also felt that way earlier in college. One graphical way is their behavior as $x \rightarrow \infty$. Draw graphs of $ e^x, e^{-x}$, also draw sum (double the average) and difference and see their asymptotic behavior. $ \cosh x$ seems to be snugly or evenly suspended between them and $ \sinh x$ is odd function like the cubic $ y= x^3$ .

Answer 3

Like their trigonometric counterparts, the cosine is even and the sine is odd and they share the value at $0$.

If you can't remember the latter, use the Euler formula

$$e^{i0}=\cos(0)+i\sin(0)=1.$$

Answer 4

This is general, if $x, -x \in \text{dom}(f)$ then we have the following identity: $$ f(x) = \underbrace{\frac{1}{2}\left( f(x)+f(-x) \right)}_{even} + \underbrace{\frac{1}{2} \left(f(x)-f(-x) \right)}_{odd}$$ when we apply this identity to the exponential function it yields the hyperbolic cosine and hyperbolic sine functions which are even/odd in the same fashion as sine an cosine: $$ e^x = \underbrace{\frac{1}{2}\left( e^x+e^{-x} \right)}_{\cosh(x)} + \underbrace{\frac{1}{2} \left(e^x-e^{-x} \right)}_{\sinh(x)} $$ The reason for the term hyperbolic is that these functions parametrize hyperbolas in the same way ordinary trigonometric functions parametrize circles: $$ x = \cosh t, y = \sinh t \ \ \Rightarrow \ \ x^2-y^2= \cosh^2t - \sinh^2 t = 1.$$ But, this is only a parametrization of $x>0$ branch of the hyperbola... as the story goes, cosh and sinh are similar to cosine and sine, but, not the same.

Answer 5

Remember sinh(x) is "smaller" than cosh(x)!

Answer 6

This approach always worked for me.