Learning Objectives:

__Math Tutoring on Using the Properties of Hyperbolic ____Functions__

**Hyperbolic functions** are written like the **trig functions cos, sin, tan,** etc., but have an **'h'** at the end, such as **cosh(x), sinh(x), and tanh(x). **The **hyperbolic function f(x) = cosh x** is defined by the formula **cosh x =(eˣ + e¯ˣ) / 2. **The function satisfies the conditions **cosh 0 = 1 **and **cosh x = cosh(−x). **The **hyperbolic function f(x) = sinh x** is defined by the formula **sinh x = (eˣ - e¯ˣ) / 2.** The function satisfies the conditions **sinh 0 = 0 **and **sinh(−x) = − sinh x. **For large values of **x** the graphs of **sinh x **and **cosh x** are close together. For large negative values of **x** the graphs of **sinh x and − cosh x** are close together.

Therefore, **tanh x = sinh x / cosh x = (eˣ - e¯ˣ) / (eˣ + e¯ˣ).**

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**Hook Questions:**

*1. **What is hyperbolic sine?*

*2. **What is Coshx?*

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