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.
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 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.
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.
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.
It is defined as the ratio of the hyperbolic sine to the hyperbolic cosine.
tanh x = sinh x / cosh x.
It is defined as the reciprocal of the hyperbolic sine.
cosech x = 1 / sinh x.
It is defined as the reciprocal of the hyperbolic cosine.
sech x = 1 / cosh x.
It is defined as the reciprocal of the hyperbolic tangent.
coth x = 1 / tanh x.
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.
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.
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.
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 -.
