Just as there are relationships between the trig functions that we can use to rewrite equations, to simplify them or to get more information out, so there are similar relationships between the hyperbolic functions.
In fact you might be surprised HOW similar!
The most commonly-used trig identity is this one:
cos2(q)+sin2(q)=1.
The corresponding hyperbolic identity is this:
cosh2(x)-sinh2(x)=1.
Notice that there's now a MINUS sign in the middle instead of a PLUS. (Also, I've changed from using "q" as an argument to using "x", because in many problems the trig functions often have an angle as argument, while the hyperbolic functions generally don't).
The point is this:
For every trig identity there is a corresponding hyperbolic identity
and they are very similar in form, with sin replaced by sinh and cos
replaced by cosh and so on.
However, they are not exactly the same, there is one important difference: whenever you have a product of two sines, then you replace it by MINUS a product of two sinh's. For example a sin2 term, (that's sin-squared) is replaced by -sinh2.
That's why in the identity I gave above the sign in the middle had changed from plus to minus.
This rule is called Osborn's rule, and you have to be a bit careful when using it. That's because sometimes the sin2 term is "hidden".
For example, remember the trig identity 1+tan2(q) = sec2(q).
Well, the corresponding hyperbolic identity is 1-tanh2(x) = sech2(x).
So the sign has changed, and that's because tan is really sin/cos, so the tan2 contains a hidden sin2 within it.
Here's a table of the most common identities, both the trig versions and the hyperbolic versions.
| Trig identity | Hyperbolic identity |
|---|---|
| cos2(x)+ sin2(x)=1 | cosh2(x)- sinh2(x)=1 |
| sin(2x)=2sin(x)cos(x) | sinh(2x)=2sinh(x)cosh(x) |
| cos(2x)=cos2(x)-sin2(x) | cosh(2x)=cosh2(x)+sinh2(x) |
| tan(2x)=2tan(x)/(1-tan2(x)) | tanh(2x)=2tanh(x)/(1+tanh2(x)) |
| sin(A)+sin(B)=2sin[(A+B)/2]cos[(A-B)/2] | sinh(A)+sinh(B)=2sinh[(A+B)/2]cosh[(A-B)/2] |
| sin(A)-sin(B)=2cos[(A+B)/2]sin[(A-B)/2] | sinh(A)-sinh(B)=2cosh[(A+B)/2]sinh[(A-B)/2] |
| cos(A)+cos(B)=2cos[(A+B)/2]cos[(A-B)/2] | cosh(A)+cosh(B)=2cosh[(A+B)/2]cosh[(A-B)/2] |
| cos(A)-cos(B)=-2sin[(A+B)/2]sin[(A-B)/2] | cosh(A)-cosh(B)=2sinh[(A+B)/2]sinh[(A-B)/2] |
| 1+tan2(x)=sec2(x) | 1-tanh2(x)=sech2(x) |
Notice that the sign has changed in the identities in rows 1, 3, 4, 8 and 9 in the above table, as each involves a product of sines.