Solving hyperbolic equations using hyperbolic identities

Understanding Hyperbolic Identities

The hyperbolic functions sinh, cosh and tanh are analogues of the standard trigonometric functions sin, cos and tan.
Hyperbolic functions, can be defined through Euler’s formula and have properties that mirror those of their trigonometric counterparts.
The hyperbolic identities often used in solving hyperbolic equations include:
- cosh^2 x - sinh^2 x = 1
- tanh^2 x + sech^2 x = 1
- sinh 2x = 2sinh x cosh x
- cosh 2x = cosh^2 x + sinh^2 x

Solving Hyperbolic Equations

One method to solve hyperbolic equations is to isolate the hyperbolic function and then compare it to known values.
For example, to solve sinh x = √3, you would isolate sinh on one side of the equation and compare √3 to values of sinh that you know from the hyperbolic function graph.
In other cases, hyperbolic equations may be solved by using a change of variable or by employing the hyperbolic identities to simplify the equation.

Applying Hyperbolic Identities

Hyperbolic identities are useful in simplifying equations so as to make it easier to isolate the hyperbolic function, or in expressing one hyperbolic function in terms of another.
For instance, an equation of the form cosh x = sinh x + √(sinh^2 x + 1) simplifies immediately by using the identity cosh^2 x - sinh^2 x = 1, which allows us to cancel out the √(sinh^2 x + 1) term on the right side.
Hyperbolic identities may also be used to express one hyperbolic function in terms of another. This is often necessary when dealing with composite hyperbolic functions like cosh (sinh x).

A proper understanding of these concepts will go a long way in making the process of solving hyperbolic equations using hyperbolic identities less daunting.