The image asks to "Define Hyperbolic Functions." Let's go over that.

Hyperbolic functions are analogs of trigonometric functions but are related to hyperbolas instead of circles. The two primary hyperbolic functions are the hyperbolic sine ($\sinh x$) and the hyperbolic cosine ($\cosh x$), and they are defined using exponential functions:

1. Hyperbolic sine: $\sinh x = \frac{e^x - e^{-x}}{2}$
2. Hyperbolic cosine: $\cosh x = \frac{e^x + e^{-x}}{2}$

From these, we can derive other hyperbolic functions, just like in trigonometry:

• Hyperbolic tangent: $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
• Hyperbolic cotangent: $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$
• Hyperbolic secant: $\text{sech } x = \frac{1}{\cosh x}$
• Hyperbolic cosecant: $\text{csch } x = \frac{1}{\sinh x}$

These functions have important properties and are used in various branches of mathematics, including calculus and differential equations. Unlike their trigonometric counterparts, they are not periodic, and their graphs resemble exponential growth and decay.

Do you need further explanations or specific details about any of these functions?

Here are some related questions:

1. How are hyperbolic functions used in solving differential equations?
2. What are the derivatives of hyperbolic functions?
3. What real-world applications involve hyperbolic functions?
4. How do hyperbolic functions relate to trigonometric functions geometrically?
5. What are the inverses of hyperbolic functions?

Tip: Hyperbolic functions are useful for describing systems that exhibit exponential growth or decay, such as in physics and engineering problems involving hyperbolic trajectories.