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Questions and Answers
What is another notation for the area hyperbolic sine function?
What is another notation for the area hyperbolic sine function?
What is the real argument called that the hyperbolic functions take?
What is the real argument called that the hyperbolic functions take?
What is another notation for the area hyperbolic secant function?
What is another notation for the area hyperbolic secant function?
What is the size of a hyperbolic angle in relation to its sector?
What is the size of a hyperbolic angle in relation to its sector?
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What is another notation for the area hyperbolic cotangent function?
What is another notation for the area hyperbolic cotangent function?
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Study Notes
Hyperbolic Functions
- Hyperbolic functions are analogues of trigonometric functions, but defined using the hyperbola instead of the circle.
- The points (cosh t, sinh t) form the right half of the unit hyperbola, similar to how (cos t, sin t) form a circle with a unit radius.
- The derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively, analogous to the derivatives of sin(t) and cos(t).
Applications of Hyperbolic Functions
- Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.
- They occur in the solutions of many linear differential equations, such as the equation defining a catenary.
- They also occur in the solutions of cubic equations and Laplace's equation in Cartesian coordinates.
- Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
Basic Hyperbolic Functions
- The basic hyperbolic functions are: hyperbolic sine "sinh" (), hyperbolic cosine "cosh" ().
- The derived hyperbolic functions are: hyperbolic tangent "tanh" (), hyperbolic cotangent "coth" (), hyperbolic secant "sech" (), hyperbolic cosecant "csch" or "cosech" ().
- These derived hyperbolic functions correspond to the derived trigonometric functions.
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Description
Test your knowledge of hyperbolic functions, which are analogues of trigonometric functions but defined using the hyperbola. Explore their properties, derivatives, and applications.
