Arccosh

Definition

arccosh is the inverse hyperbolic cosine function, denoted as \(\text{arccosh}(x)\) or \(\cosh^{-1}(x)\). It takes a real number \(x\) (where \(x \geq 1\)) and returns the value of \(y\) such that \(\cosh(y) = x\). This function is crucial for understanding the properties of hyperbolic functions and their applications in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. The domain of arccosh is restricted to \([1, \infty)\), as the values of the hyperbolic cosine function start from 1 and increase.
2. The range of arccosh is \([0, \infty)\), meaning the output values will always be non-negative.
3. The derivative of arccosh can be expressed as \(\frac{d}{dx} \text{arccosh}(x) = \frac{1}{\sqrt{x^2 - 1}}\), which highlights its behavior near its domain limits.
4. To compute arccosh using natural logarithms, the formula is given by \(\text{arccosh}(x) = \ln(x + \sqrt{x^2 - 1})\).
5. Graphically, the function arccosh has a vertical asymptote at x = 1 and increases without bound as x increases.

Review Questions

• Explain how arccosh relates to the hyperbolic cosine function and why it's defined only for values greater than or equal to 1.
  • arccosh is the inverse function of the hyperbolic cosine function, which means it reverses the effect of cosh. Since cosh(x) produces outputs starting at 1 and going to infinity as x increases, arccosh is only defined for values greater than or equal to 1. For any input less than 1, there is no corresponding y such that cosh(y) equals that input, making those inputs invalid for arccosh.
• Demonstrate how to calculate arccosh using both its definition and its logarithmic formula.
  • To calculate arccosh for a given value, you can either use its definition or apply the logarithmic formula. Using the definition, if you want to find arccosh(2), you look for y such that cosh(y) = 2. Alternatively, you can use the logarithmic approach: apply the formula \(\text{arccosh}(2) = \ln(2 + \sqrt{2^2 - 1}) = \ln(2 + \sqrt{3})\). Both methods yield the same result.
• Evaluate how understanding arccosh impacts your comprehension of other mathematical concepts involving hyperbolic functions and their applications.
  • Understanding arccosh is essential for grasping more complex mathematical concepts involving hyperbolic functions since it provides insight into how these functions behave under inversion. This knowledge is particularly useful in fields like calculus, physics, and engineering where hyperbolic functions model real-world phenomena. For example, knowing how to manipulate arccosh helps solve problems related to hyperbolic geometry and can aid in analyzing systems with hyperbolic characteristics, thereby enhancing your overall problem-solving toolkit.

Related terms

cosh: The hyperbolic cosine function, defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\), which relates to arccosh as its inverse.

hyperbolic functions: A set of functions including sinh, cosh, tanh, and their inverses, which are analogous to trigonometric functions but based on hyperbolas.

exponential function: A mathematical function defined as \(e^x\), which is foundational for defining hyperbolic functions, including cosh and arccosh.