Hyperbolic trigonometric functions

The hyperbolic trigonometric functions, also referred to as simply "hyperbolic functions," are analogous to the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle. This has the effect of removing any i's that appear in the complex definition of the standard trigonometric functions. As such, they tend to differentiate in an analogous way to standard trigonometric functions, up to perhaps a negative sign. They are defined as:

• Hyperbolic sine: $\sinh(x)={\frac {e^{x}-e^{-x}}{2}}$
• Hyperbolic cosine: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh(x) = \frac{e^x + e^{-x}}{2}}
• Hyperbolic tangent: $\tanh(x)={\frac {\sinh(x)}{\cosh(x)}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$
• Hyperbolic cosecant: ${\text{csch}}(x)={\frac {1}{\sinh(x)}}={\frac {2}{e^{x}-e^{-x}}}$
• Hyperbolic secant: ${\text{sech}}(x)={\frac {1}{\cosh(x)}}={\frac {2}{e^{x}+e^{-x}}}$
• Hyperbolic cotangent: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}}

Graphs

sinh and cosh	tanh and coth

Identities

The hyperbolic trigonometric functions have many identities that are similar to those of trigonometric functions. These can be remembered by replacing instances of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2{x}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\sinh^2{x}} . For example, the trigonometric identity,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2{x}+\sin^2{x} = 1 }

has the corresponding hyperbolic identity,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh^2{x}-\sinh^2{x} = 1 }

Other identities include:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh{2x} =2\sinh{x}\cosh{x} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh{2x} = \cosh^2{x}+\sinh^2{x} }

Also, note that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh{iz} = i\sin{z} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh{iz}=\cos{z} }

See also

• Trigonometric function