Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4

Verified step by step guidance

1. Recall the definition of the hyperbolic cotangent function: \(\text{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\).
2. Express \(\cosh(x)\) and \(\sinh(x)\) in terms of exponential functions: \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
3. Substitute \(x = 4\) into these expressions to find \(\cosh(4)\) and \(\sinh(4)\).
4. Calculate the values of \(\cosh(4)\) and \(\sinh(4)\) using a calculator or computational tool.
5. Divide \(\cosh(4)\) by \(\sinh(4)\) to find \(\text{coth}(4)\) and round the result to four decimal places.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh, cosh, and coth, are analogs of trigonometric functions but based on hyperbolas. The function coth(x) is defined as cosh(x) divided by sinh(x), and it is important to understand their definitions to evaluate expressions correctly.

Domain and Existence of Hyperbolic Functions

The domain of hyperbolic functions like coth(x) excludes values where the denominator is zero. Since sinh(x) = 0 at x = 0, coth(x) is undefined there. For x = 4, sinh(4) ≠ 0, so coth(4) exists and can be evaluated.

Using Calculators for Hyperbolic Functions

Modern calculators often have built-in hyperbolic function keys (sinh, cosh, tanh). To find coth(x), compute cosh(x) and sinh(x) separately and divide. Reporting answers to four decimal places requires rounding the calculator output accordingly.