2.5.1: Resources and Key Concepts

Resources

Prerequisite Topics

The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Solving Equations
  • Solving Exponential Equations: Implicit in finding inverse hyperbolic functions, as it involves solving for \(y\) in expressions like \(x = \frac{e^y - e^{-y}}{2}\).
  • Solving Logarithmic Equations: Inverse hyperbolic functions are defined in terms of natural logarithms.
• Exponential Functions
  • Graphing Exponential Functions: Understanding the graphs of \(e^x\) and \(e^{-x}\) helps in visualizing the graphs of hyperbolic functions.
• Logarithmic Functions
  • Laws of Logarithms: Used in simplifying expressions involving hyperbolic functions when written in exponential form, particularly after applying logarithms (Example 1).
• Trigonometry
  • Proving Trigonometric Identities: The same skills you used to prove trigonometric identities in your Trigonometry course are required in this section.
  • Sector Area: The area of the sector subtended by the angle \( \theta \) in a unit circle is our motivation to defining something similar to trigonometric functions when dealing with the unit hyperbola.

Videos

• Hyperbolic functions
  • An Introduction to Hyperbolic Functions
• Unit hyperbola
  • A Recreational Interlude: Euler’s Formula and the Unit Circle
• Hyperbolic identities
  • Hyperbolic Identities
• Inverse hyperbolic functions
  • Inverse Hyperbolic Functions
• Logarithmic form of inverse hyperbolic functions
  • Proofs in Differential Calculus - \( \text{arcosh}(y) = \ln(y + \sqrt{y^2 - 1}) \)

Key Concepts

Definitions

• Circular Functions: An alternative name for trigonometric functions, emphasizing their definition based on the unit circle.
• Unit Hyperbola: For the purposes of this material, the right branch of the hyperbola defined by the relation \(x^2 - y^2 = 1\).
• Hyperbolic Cosine (geometric interpretation): If \(s/2\) is the area of the region bounded by the positive \(x\)-axis, the unit hyperbola, and the line segment connecting the origin to the point \((x,y)\) on the hyperbola, then \(\cosh(s)\) is the \(x\)-value of this terminal point.
• Hyperbolic Sine (geometric interpretation): If \(s/2\) is the area of the region bounded by the positive \(x\)-axis, the unit hyperbola, and the line segment connecting the origin to the point \((x,y)\) on the hyperbola, then \(\sinh(s)\) is the \(y\)-value of this terminal point.
• Hyperbolic Functions (Exponential Definitions):
  • Hyperbolic Cosine: \(\cosh(t) = \frac{e^t + e^{-t}}{2}\)
  • Hyperbolic Sine: \(\sinh(t) = \frac{e^t - e^{-t}}{2}\)
  • Hyperbolic Tangent: \(\tanh(t) = \frac{\sinh(t)}{\cosh(t)} = \frac{e^t - e^{-t}}{e^t + e^{-t}}\)
  • Hyperbolic Cosecant: \(\text{csch}(t) = \frac{1}{\sinh(t)} = \frac{2}{e^t - e^{-t}}\)
  • Hyperbolic Secant: \(\text{sech}(t) = \frac{1}{\cosh(t)} = \frac{2}{e^t + e^{-t}}\)
  • Hyperbolic Cotangent: \(\coth(t) = \frac{\cosh(t)}{\sinh(t)} = \frac{e^t + e^{-t}}{e^t - e^{-t}}\)
• Catenary: The shape of a hanging chain or cable, which can be represented by the hyperbolic cosine function.

Theorems

• Theorem: Hyperbolic Functions (Exponential Forms): Lists the exponential definitions for \(\cosh(t)\), \(\sinh(t)\), \(\tanh(t)\), \(\text{csch}(t)\), \(\text{sech}(t)\), and \(\coth(t)\).
• Theorem: The Fundamental Hyperbolic Identity: \(\cosh^2(t) - \sinh^2(t) = 1\).
• Theorem: Basic Hyperbolic Identities:
  1. \(\cosh(-t) = \cosh(t)\)
  2. \(\sinh(-t) = -\sinh(t)\)
  3. \(\cosh(t) + \sinh(t) = e^t\)
  4. \(\cosh(t) - \sinh(t) = e^{-t}\)
  5. \(1 - \tanh^2(t) = \text{sech}^2(t)\)
  6. \(\coth^2(t) - 1 = \text{csch}^2(t)\)
  7. \(\sinh(t \pm v) = \sinh(t)\cosh(v) \pm \cosh(t)\sinh(v)\)
  8. \(\cosh(t \pm v) = \cosh(t)\cosh(v) \pm \sinh(t)\sinh(v)\)
• Inverse Hyperbolic Functions (Logarithmic Forms):
  • \(\text{sinh}^{-1}x = \text{arsinh } x = \ln(x + \sqrt{x^2+1})\)
  • \(\text{cosh}^{-1}x = \text{arccosh } x = \ln(x + \sqrt{x^2-1})\) for \(x \ge 1\)
  • \(\text{tanh}^{-1}x = \text{artanh } x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\) for \(|x| < 1\)
  • \(\text{coth}^{-1}x = \text{arccoth } x = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)\) for \(|x| > 1\)
  • \(\text{sech}^{-1}x = \text{arcsech } x = \ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)\) for \(0 < x \le 1\)
  • \(\text{csch}^{-1}x = \text{arccsch } x = \ln\left(\frac{1}{x} + \frac{\sqrt{1+x^2}}{|x|}\right)\) for \(x \neq 0\)

Common Mistakes

• The Argument of a Hyperbolic Function is Not an Angle, Nor an Arc Length: It is critical to point out that the argument \(s\) in \(\cosh(s)\) or \(\sinh(s)\) (when related to the unit hyperbola area interpretation) is not an angle or a direct arc length along the hyperbola in the same way \(\theta\) is for circular functions. The argument \(s\) is such that \(s/2\) is an area.
• Independent Variable as a Dummy Variable: The independent variable \(x\) in the theorem for Inverse Hyperbolic Functions is a dummy variable. It is not the \(x\)-coordinate of the terminal point of the arc of length \(t\) along the unit hyperbola.