Geometry with hyperbolic functions

In summary, hyperbolic functions, such as sinh, cosh, and tanh, have geometric interpretations and can be used in hyperbolic and elliptic coordinate systems. However, it may be difficult to understand their geometric deductions and how they can help solve geometric problems. It is suggested to study hyperbolic geometry, the hyperbola, and the exponential and logarithmic functions based on the Euler number base to gain a better understanding.
  • #1
Jhenrique
685
4
Is known that in every rectangle triangle the following relationships are true:

imagem.png


But, how use geometrically the function sinh, cosh, and tanh?
 
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  • #2
Study hyperbolic geometry
 
  • #3
  • #4
SteamKing said:
This article gives one geometric interpretation of sinh and cosh:

http://en.wikipedia.org/wiki/Hyperbolic_function

(check out the figure at the upper RHS of the article)

But I still don't understood how the hyperbolic functions help me to solve geometric problems.
 
  • #5
Jhenrique said:
But I still don't understood how the hyperbolic functions help me to solve geometric problems.

Why do you expect that they do?
 
  • #6
micromass said:
Why do you expect that they do?

Hyperbolic functions appears in hyperbolic and elliptic coordinate system, but I don't understand the geometric deduction for these formulas.
 
  • #7
Jhenrique said:
Hyperbolic functions appears in hyperbolic and elliptic coordinate system, but I don't understand the geometric deduction for these formulas.

OK. Which formulas don't you understand the deduction of? What is the deduction and what don't you understand about them?
 
  • #9
Jhenrique said:
Hyperbolic functions appears in hyperbolic and elliptic coordinate system, but I don't understand the geometric deduction for these formulas.

Jhenrique said:
But I still don't understood how the hyperbolic functions help me to solve geometric problems.

micromass said:
Study hyperbolic geometry


This last one was said best.


Trigonometry is Geometry based on a unit circle.
Hyperbolic sines and cosines are based on the unit hyperbola.
Study the hyperbola and the exponential and logarithmic functions based on the Euler number base.
 
  • #10
I already studied and I continue don't understand how use it
 

1. What are hyperbolic functions in geometry?

Hyperbolic functions are mathematical functions that are used to describe the relationship between the sides and angles of a hyperbolic triangle. They are derived from the hyperbolic sine, cosine, and tangent functions, and are commonly used in non-Euclidean geometries, such as hyperbolic geometry.

2. How are hyperbolic functions used in geometry?

In geometry, hyperbolic functions are used to calculate the lengths of the sides and angles in a hyperbolic triangle. They are also used to solve problems in hyperbolic geometry, such as finding the area of a hyperbolic polygon or the distance between two points on a hyperbolic surface.

3. What is the difference between hyperbolic and trigonometric functions?

Hyperbolic functions are similar to trigonometric functions, but they are defined for hyperbolic angles instead of circular angles. Hyperbolic functions are also related to the exponential function e^x, while trigonometric functions are related to the unit circle.

4. Can hyperbolic functions be graphed?

Yes, hyperbolic functions can be graphed. The graphs of hyperbolic functions have a similar shape to the graphs of trigonometric functions, but they are flatter and more spread out.

5. What are some real-world applications of hyperbolic functions in geometry?

Hyperbolic functions have many real-world applications in geometry, such as in the design of curved structures, such as bridges and arches. They are also used in physics and engineering to model the behavior of waves and vibrations. Additionally, hyperbolic geometry is used in the field of computer graphics to create 3D models and animations.

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