Hyperbolic Functions

Hyperbolic Functions

• Hyperbolic functions are similar to ordinary trigonometric functions, but have different properties and rules.
• There are six basic hyperbolic functions: sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), coth (hyperbolic cotangent), sech (hyperbolic secant), and csch (hyperbolic cosecant).
• The definitions of hyperbolic sine (sinh) and hyperbolic cosine (cosh) are derived from their equivalent structure in complex numbers. They can be expressed in terms of exponential functions as follows: sinh x = (e^x - e^-x) / 2, cosh x = (e^x + e^-x) / 2.
• The equation that relates the hyperbolic sine and cosine is cosh²x - sinh²x = 1, which is similar to the Pythagorean identity for trigonometric functions, except it lacks a negative sign.
• Hyperbolic tangent can be defined as the ratio of hyperbolic sine to hyperbolic cosine, tanh x = sinh x / cosh x.
• Hyperbolic functions also have reciprocal hyperbolic functions. These can be defined as: coth x = 1 / tanh x, sech x = 1 / cosh x, csch x = 1 / sinh x.
• Graphs of hyperbolic functions are distinct from their trigonometric counterparts, illustrating their different properties.
• The derivatives of hyperbolic functions can be expressed in terms of other hyperbolic functions. For example, the derivative of sinh x is cosh x and the derivative of cosh x is sinh x.
• Hyperbolic functions can be used to solve specific types of differential equations, and have applications in the fields of engineering, mathematics, and physics.
• Integration of hyperbolic functions is a crucial part to understand and can be achieved through standard integration techniques along with the use of identities and properties.
• Hyperbolic functions are useful in manipulation and simplification of expressions, particularly when dealing with exponentials and logarithms.

Spend sufficient time understanding the characteristics, rules, and applications of hyperbolic functions for thorough preparation.