# Hyperbolic Functions vs. Trigonometric Functions

## What's the Difference?

Hyperbolic functions and trigonometric functions are both types of mathematical functions that are used to describe relationships between angles and sides in triangles. However, hyperbolic functions are based on the hyperbola, while trigonometric functions are based on the unit circle. Hyperbolic functions are defined in terms of exponentials, while trigonometric functions are defined in terms of ratios of sides in a right triangle. Additionally, hyperbolic functions have their own set of identities and properties that are distinct from trigonometric functions. Overall, while both types of functions are used in mathematics to model various phenomena, they have different origins and properties that make them unique.

## Comparison

Attribute | Hyperbolic Functions | Trigonometric Functions |
---|---|---|

Definition | Defined in terms of exponential functions | Defined in terms of ratios of sides of a right triangle |

Range of Input | Real numbers | Angles |

Notation | sinh, cosh, tanh, etc. | sin, cos, tan, etc. |

Relationship | Related to the unit hyperbola | Related to the unit circle |

Identities | Hyperbolic identities | Trigonometric identities |

## Further Detail

### Introduction

Hyperbolic functions and trigonometric functions are two important classes of mathematical functions that are widely used in various fields of mathematics and science. While they may seem similar at first glance, there are some key differences between the two types of functions that make them unique and useful in different contexts.

### Definition

Trigonometric functions, such as sine, cosine, and tangent, are defined in terms of the ratios of the sides of a right triangle. These functions are periodic, meaning they repeat their values at regular intervals. Hyperbolic functions, on the other hand, are defined in terms of the hyperbola, a curve that is similar to the ellipse but opens up in two directions. Hyperbolic functions are not periodic and have different properties compared to trigonometric functions.

### Graphs

When graphed, trigonometric functions produce sinusoidal waves that repeat themselves over a certain period. The graphs of sine and cosine functions are smooth and continuous, with a range of -1 to 1. In contrast, hyperbolic functions produce hyperbolas when graphed, which are symmetrical curves that open up in two directions. The graphs of hyperbolic functions are also smooth and continuous, but they do not repeat themselves like trigonometric functions.

### Relationships

Trigonometric functions are closely related to the unit circle, where the values of sine and cosine correspond to the y and x coordinates of a point on the circle. These functions are used to describe periodic phenomena such as waves and oscillations. Hyperbolic functions, on the other hand, are used to describe exponential growth and decay processes, as well as other non-periodic phenomena. The relationships between hyperbolic functions are different from those of trigonometric functions, reflecting their unique properties.

### Identities

Trigonometric functions have a set of identities that relate different trigonometric functions to each other. These identities are used to simplify trigonometric expressions and solve trigonometric equations. Hyperbolic functions also have their own set of identities, known as hyperbolic identities, which relate different hyperbolic functions to each other. These identities are used in a similar way to trigonometric identities, but they apply to hyperbolic functions instead.

### Applications

Trigonometric functions are used in a wide range of applications, including physics, engineering, and signal processing. They are used to model periodic phenomena such as sound waves, light waves, and mechanical vibrations. Hyperbolic functions are also used in various fields, such as physics, engineering, and mathematics. They are used to model exponential growth and decay processes, as well as other non-periodic phenomena.

### Integration and Differentiation

Trigonometric functions have well-known derivatives and integrals that are used in calculus to solve a variety of problems. The derivatives of trigonometric functions are used to find the rate of change of a function, while the integrals of trigonometric functions are used to find the area under a curve. Hyperbolic functions also have derivatives and integrals that are used in calculus, but they are different from those of trigonometric functions due to the unique properties of hyperbolic functions.

### Conclusion

In conclusion, hyperbolic functions and trigonometric functions are two important classes of mathematical functions that have distinct properties and applications. While trigonometric functions are used to model periodic phenomena, hyperbolic functions are used to model exponential growth and decay processes. Both types of functions have their own set of identities, derivatives, and integrals that are used in various fields of mathematics and science. Understanding the differences between hyperbolic functions and trigonometric functions is essential for using them effectively in different contexts.

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