Table of Contents

Hyperbolic Functions Formulas and Identities

Definitions of Hyperbolic Functions

• Hyperbolic sine : \( \sinh x = \dfrac{e^x - e^{-x}}{2} \)
• Hyperbolic cosine : \( \cosh x = \dfrac{e^x + e^{-x}}{2} \)
• Hyperbolic tangent : \( \tanh x = \dfrac{\sinh x}{\cosh x} \)
• Hyperbolic cotangent : \( \coth x = \dfrac{1}{\tanh x} = \dfrac{\cosh x}{\sinh x} \)
• Hyperbolic secant : \( \sech x = \dfrac{1}{\cosh x} \)
• Hyperbolic cosecant : \( \csch x = \dfrac{1}{\sinh x} \)

Negative Argument Formulas

• \( \sinh (-x) = - \sinh x \) , odd function , graph symmetric with respect to the origin.
• \( \cosh (-x) = \cosh x \) , even function , graph symmetric with \( y \) axis.
• \( \tanh (-x) = - \tanh x \) , odd function , graph symmetric with respect to the origin.

Hyperbolic Functions of Sums/Difference

• \( \cosh(A \pm B) = \cosh A \cosh B \pm \sinh A \sinh B \)
• \( \sinh(A \pm B) = \sinh A \cosh B \pm \cosh A \sinh B \)
• \( \tanh(A \pm B) = \dfrac{\tanh A \pm \tanh B}{1 \pm \tanh A \tanh B} \)

Sums of Hyperbolic Functions to Product of Hyperbolic Functions

• \( \sinh A + \sinh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right) \)
• \( \sinh A - \sinh B = 2 \sinh \left( \dfrac {A - B}{2} \right) \cosh \left( \dfrac {A + B}{2} \right) \)
• \( \cosh A + \cosh B = 2 \cosh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right) \)
• \( \cosh A - \cosh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \sinh \left( \dfrac {A-B}{2} \right) \)

Product of Hyperbolic Functions to Sum of Hyperbolic Functions

• \( \sinh A \cosh B = \dfrac{1}{2} \; [ \sinh(A+B) + \sinh(A-B) ] \)
• \( \cosh A \cosh B = \dfrac{1}{2} \; [ \cosh(A+B) + \cosh(A-B) ] \)
• \( \sinh A \sinh B = \dfrac{1}{2} \; [ \cosh(A+B) - \cosh(A-B) ] \)

Hyperbolic Functions of Multiple Arguments

• \( \sinh (2 A) = 2 \sinh A \cosh A \)
• \( \cosh (2 A) = 2 \sinh^2 A + 1 = 2 \cosh^2 A -1 \)
• \( \sinh (3 A) = 4 \sinh^3 A + 3 \sinh A \)
• \( \cosh (3 A) = 4\cosh^3 A - 3 \cosh A \)

Relations Between Hyperbolic Functions

• \( \cosh^2 A - \sinh^2 A = 1 \)
• \( \tanh^2 A + \sech^2 A = 1 \)
• \( \coth^2 A - \csch^2 A = 1 \)

Power Reducing Identities

• \( \sinh^2 A = \dfrac{1}{2} [\cosh (2A) - 1] \)
• \( \cosh^2 A = \dfrac{1}{2} [\cosh (2A) + 1] \)
• \( \sinh^3 A = \dfrac{\sinh(3A) - 3 \sinh A}{4} \)
• \( \cosh^3 A = \dfrac{\cosh(3A) + 3 \cosh A}{4} \)
• \( \sinh^4 A = \dfrac{\cosh (4A) - 4 \cosh (2A) + 3}{8} \)
• \( \cosh^4 A = \dfrac{\cosh (4A) + 4 \cosh (2A) + 3}{8} \)
• \( \sinh^5 A = \dfrac{\sinh (5A) - 5 \sinh (3A) + 10 \sinh A}{16} \)
• \( \cosh^5 A = \dfrac{\cosh (5A) + 5 \cosh (3A) + 10 \cosh A}{16} \)

Inverse Hyperbolic Functions

• \( \sinh^{-1} A = \ln (x+\sqrt{x^2+1}) \)
• \( \cosh^{-1} A = \ln (x+\sqrt{x^2-1}) \; , \; x \ge 1\)
• \( \tanh^{-1} A = \dfrac{1}{2}\ln \left(\dfrac{1+x}{1-x} \right) \; , \; -1 \lt x \lt 1\)
• \( \csch^{-1} A = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} + 1} \right) \; , \; x \ne 0\)
• \( \sech^{-1} A = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} - 1} \right) \; , \; 0 \lt x \le 1\)
• \( \coth^{-1} A = \dfrac{1}{2}\ln \left(\dfrac{x+1}{x-1} \right) \; , \; |x| \gt 1\)

Derivative of Hyperbolic Functions and Their Inverses

Indefinite Integrals of Hyperbolic Functions

More References and Links