Certainly! The hyperbolic functions can be defined in terms of the exponential function $e^x$ as follows:

1. Hyperbolic Sine ($\sinh(x)$):

$\sinh(x) = \frac{e^x - e^{-x}}{2}$ This function is analogous to the sine function but is defined in terms of exponentials.

2. Hyperbolic Cosine ($\cosh(x)$):

$\cosh(x) = \frac{e^x + e^{-x}}{2}$ Similar to the cosine function, but using exponentials.

3. Hyperbolic Tangent ($\tanh(x)$):

$\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ This is the ratio of hyperbolic sine and hyperbolic cosine.

4. Hyperbolic Cotangent ($\coth(x)$):

$\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ The reciprocal of hyperbolic tangent.

5. Hyperbolic Secant ($\sech(x)$):

$\sech(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}$ The reciprocal of hyperbolic cosine.

6. Hyperbolic Cosecant ($\csch(x)$):

$\csch(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}}$ The reciprocal of hyperbolic sine.

These definitions provide the connection between hyperbolic functions and exponential functions, forming the basis for their properties and behavior.

Do you want details on the properties or uses of these functions?

Related questions:

1. What are the key differences between hyperbolic and trigonometric functions?
2. How do the graphs of hyperbolic sine and cosine look compared to their trigonometric counterparts?
3. Can you explain the relationship between the hyperbolic identities and the exponential function?
4. How are hyperbolic functions used in solving differential equations?
5. What real-world applications make use of hyperbolic functions?

Tip:

The hyperbolic identity $\cosh^2(x) - \sinh^2(x) = 1$ mirrors the trigonometric identity $\cos^2(x) + \sin^2(x) = 1$, but with a subtle difference in signs.