Hyperbolic Functions Study Notes

Hyperbolic Functions

Definition of Hyperbolic Functions

• Hyperbolic Functions are analogs of trigonometric functions but for hyperbolas.

• Key hyperbolic functions include:

• sinh (hyperbolic sine): Defined as
  [ sinh(x) = \frac{e^x - e^{-x}}{2} ]

• cosh (hyperbolic cosine): Defined as
  [ cosh(x) = \frac{e^x + e^{-x}}{2} ]

• tanh (hyperbolic tangent): Defined as
  [ tanh(x) = \frac{sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} ]

Properties of Hyperbolic Functions

• Symmetry:

• sinh is an odd function: [ sinh(-x) = -sinh(x) ]

• cosh is an even function: [ cosh(-x) = cosh(x) ]

• Identities:

• Pythagorean identity:
  [ cosh^2(x) - sinh^2(x) = 1 ]

• Addition formulas:
  [ sinh(a + b) = sinh(a)cosh(b) + cosh(a)sinh(b) ]
  [ cosh(a + b) = cosh(a)cosh(b) + sinh(a)sinh(b) ]

Graphs of Hyperbolic Functions

• The graphs of hyperbolic functions resemble those of trigonometric functions but reflect the nature of hyperbolas.

• sinh:

• A curve that increases without bounds in both the positive and negative directions.

• Passes through the origin (0,0).

• cosh:

• A U-shaped curve that has a minimum value at (0,1).

• Functions grow exponentially for large values of x.

Applications of Hyperbolic Functions

• Used in calculations involving hyperbolic geometry, engineering (e.g., hanging cables), and physics.

• Model phenomena such as population growth, heat conduction, or in certain types of wave equations.

Key Figures

• Figures associated with hyperbolic functions from slides 7.11 to 7.21 illustrate the characteristics, graphs, and relationships of these functions.

• Important to analyze and reference these figures for a visual understanding of the hyperbolic functions' behaviors.

Conclusion

• Understanding hyperbolic functions is crucial not just for mathematics, but also for various applications in science and engineering.

• Review their properties, identities, and graphs systematically to prepare for exams or practical applications.

Mathematical Formulas of Hyperbolic Functions

Definitions

• sinh(x):

   [ sinh(x) = \frac{e^x - e^{-x}}{2} ]

• cosh(x):

   [ cosh(x) = \frac{e^x + e^{-x}}{2} ]

• tanh(x):

   [ tanh(x) = \frac{sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} ]

Identities

• Pythagorean Identity:

   [ cosh^2(x) - sinh^2(x) = 1 ]

• Addition Formulas:

   [ sinh(a + b) = sinh(a)cosh(b) + cosh(a)sinh(b) ]

   [ cosh(a + b) = cosh(a)cosh(b) + sinh(a)sinh(b) ]