Hyperbolic Functions

Definitions of Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric (circular) functions but are defined using the hyperbola rather than the circle. They are defined in terms of the exponential function $e^x$:

• Hyperbolic Sine: $\sinh x = \frac{e^x - e^{-x}}{2}$
• Hyperbolic Cosine: $\cosh x = \frac{e^x + e^{-x}}{2}$
• Hyperbolic Tangent: $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
• Hyperbolic Cosecant: $\text{csch } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$ (for $x \neq 0$)
• Hyperbolic Secant: $\text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$
• Hyperbolic Cotangent: $\coth x = \frac{1}{\tanh x} = \\frac{e^x + e^{-x}}{e^x - e^{-x}}$ (for $x \neq 0$)

Note: Unlike $\sin x$ and $\cos x$, $\sinh x$ and $\cosh x$ are not bounded. $\cosh x \ge 1$ for all real $x$.

Key Hyperbolic Identities

Similar to trigonometric identities, there are identities relating hyperbolic functions:

• $\cosh^2 x - \sinh^2 x = 1$ (Compare with $\cos^2 \theta + \sin^2 \theta = 1$)
• $1 - \tanh^2 x = \text{sech}^2 x$ (Compare with $1 + \tan^2 \theta = \sec^2 \theta$)
• $\coth^2 x - 1 = \text{csch}^2 x$ (Compare with $\cot^2 \theta + 1 = \csc^2 \theta$)
• Addition Formulas:
  • $\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$
  • $\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$
  • $\tanh(x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}$
• Double Angle Formulas:
  • $\sinh 2x = 2 \sinh x \cosh x$
  • $\cosh 2x = \cosh^2 x + \sinh^2 x = 2 \cosh^2 x - 1 = 1 + 2 \sinh^2 x$
  • $\tanh 2x = \frac{2 \tanh x}{1 + \tanh^2 x}$

Relationship Between Trigonometric and Hyperbolic Functions (Osborn's Rule)

There's a strong connection between trigonometric and hyperbolic functions when considering complex numbers ($i^2 = -1$). Euler's formula states $e^{i\theta} = \cos \theta + i \sin \theta$. Using this, we can derive:

• $\cosh(ix) = \frac{e^{ix} + e^{-ix}}{2} = \cos x$
• $\sinh(ix) = \frac{e^{ix} - e^{-ix}}{2} = i \sin x$
• $\cos(ix) = \frac{e^{i(ix)} + e^{-i(ix)}}{2} = \frac{e^{-x} + e^{x}}{2} = \cosh x$
• $\sin(ix) = \frac{e^{i(ix)} - e^{-i(ix)}}{2i} = \frac{e^{-x} - e^{x}}{2i} = \frac{-(e^x - e^{-x})}{2i} = i \sinh x$

Osborn's Rule provides a simple way to convert trigonometric identities involving $\sin$ and $\cos$ into corresponding hyperbolic identities:

1. Replace every trigonometric function with its corresponding hyperbolic function.
2. Change the sign of any term that contains a product of two sine functions (e.g., $\sin A \sin B$ or $\sin^2 A$).

Example: Start with $\cos(A+B) = \cos A \cos B - \sin A \sin B$.

1. Replace trig with hyperbolic: $\cosh(A+B) = \cosh A \cosh B - \sinh A \sinh B$.
2. Change sign of the $\sinh A \sinh B$ term (product of two sines): $\cosh(A+B) = \cosh A \cosh B + \sinh A \sinh B$. This is the correct hyperbolic identity.

Example 2: Start with $1 + \tan^2 A = \sec^2 A$. Since $\tan^2 A = \frac{\sin^2 A}{\cos^2 A}$, this implicitly contains a product of two sines.

1. Replace trig with hyperbolic: $1 + \\tanh^2 A = \\text{sech}^2 A$.
2. Change sign of the $\tanh^2 A$ term: $1 - \\tanh^2 A = \\text{sech}^2 A$. This is the correct hyperbolic identity.