AI Hyperbolic Functions Solver
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Hyperbolic functions, Identities

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AI Hyperbolic Functions Solver

The AI hyperbolic solver has the ability to solve various hyperbolic functions with easy-to-follow worksheet that includes hyperbolic, such as sinh x, cosh x, tanh x, cosech x, sech x, and coth x.

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About MathCrave AI Hyperbolic Solver

The AI hyperbolic solver is capable of solving hyperbolic functions, such as sinh x, cosh x, tanh x, cosech x, sech x, and coth x. It also provides a clear and detailed step-by-step worksheet including hyperbolic identities.

Math Problems AI Hyperbolic Functions Solver Solves

  • Sinh x, cosh x, tanh x, cosech x, sech x and coth x

  • Evaluate hyperbolic functions

  • State and proof Osborne’s rule

  • Simple hyperbolic identities proofs

  • Equations involving hyperbolic functions

  • Derivatives the series expansions for cosh x

  • Derivatives of the series expansions for sinh x

Hyperbolic Functions

Hyperbolic functions are a set of mathematical functions that are related to the exponential function. The main hyperbolic functions are sinh x, cosh x, tanh x, cosech x, sech x, and coth x. They have properties and identities similar to those of trigonometric functions but are defined over the hyperbola instead of the unit circle. Hyperbolic functions are widely used in areas such as calculus, differential equations, and mathematical physics.

Sinh x, Cosh x, Tanh x, Cosech x, Sech x, and Coth x:

Sinh x (Hyperbolic Sine)

It is defined as the ratio of the difference between the exponential of x and the exponential of -x to 2

  • sinh x = (e^x - e^(-x))/2.

Cosh x (Hyperbolic Cosine)

It is defined as the ratio of the sum of the exponential of x and -x to 2.

  • cosh x = (e^x + e^(-x))/2.

Tanh x (Hyperbolic Tangent)

It is defined as the ratio of the hyperbolic sine to the hyperbolic cosine.

  • tanh x = sinh x / cosh x.

Cosech x (Hyperbolic Cosecant)

It is defined as the reciprocal of the hyperbolic sine.

  • cosech x = 1 / sinh x.

Sech x (Hyperbolic Secant)

It is defined as the reciprocal of the hyperbolic cosine.

  • sech x = 1 / cosh x.

Coth x (Hyperbolic Cotangent)

It is defined as the reciprocal of the hyperbolic tangent.

  • coth x = 1 / tanh x.

Evaluating Hyperbolic Functions:

Hyperbolic functions can be evaluated using their respective definitions or by using exponentials. Trigonometric identities or properties can also be used to evaluate hyperbolic functions.

Simple Hyperbolic Identities Proofs

There are various identities involving hyperbolic functions that can be proved using their definitions and properties. Some examples include proving that cosh^2 x - sinh^2 x = 1 and proving that cosh(2x) = cosh^2 x + sinh^2 x.

Solving Equations Involving Hyperbolic Functions

Equations involving hyperbolic functions can be solved using algebraic manipulation, substitution, or by using specific properties of hyperbolic functions. These equations can have single or multiple solutions depending on the given conditions.

Osborne’s Rule of Hyperbolic Functions

Using Osborne's rule, which states that the six trigonometric ratios used in trigonometrical identities involving general angles can be replaced by their corresponding hyperbolic functions, with the exception that the sign of any direct or implied product of two sines must be changed.

For instance, applying Osborne's rule to the equation cos^2 x + sin^2 x = 1, we get ch^2 x - sh^2 x = 1. This means that the trigonometric functions have been replaced with their corresponding hyperbolic functions, and since sin 2 x is a product of two sines, the sign changes from + to -.

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