Problem 20 - Precalculus with Calculus Previews

Question

Find the exact numerical value of the given quantity. $$ \cosh (-\ln 3) $$

Step-by-Step Solution

Answer

\( \cosh(-\ln 3) = \frac{5}{3} \)

Step 1: Understand the Hyperbolic Cosine Function

The hyperbolic cosine function, defined as \( \cosh(x) \), is given by the formula \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). This will be useful in computing \( \cosh(-\ln 3) \).

Step 2: Apply the Formula

Substitute \( x = -\ln 3 \) into the hyperbolic cosine formula \( \cosh(-\ln 3) = \frac{e^{-\ln 3} + e^{\ln 3}}{2} \).

Step 3: Simplify the Exponential Expressions

By properties of exponents, \( e^{-\ln 3} = \frac{1}{3} \) and \( e^{\ln 3} = 3 \), because \(e^{-\ln a} = \frac{1}{a} \) and \(e^{\ln a} = a\).

Step 4: Evaluate the Expression

Substituting these values back, we get \( \cosh(-\ln 3) = \frac{\frac{1}{3} + 3}{2} = \frac{\frac{1}{3} + \frac{9}{3}}{2} = \frac{\frac{10}{3}}{2} = \frac{10}{6} = \frac{5}{3} \).

Key Concepts

Understanding the Hyperbolic Cosine

The hyperbolic cosine function, denoted as \(\cosh(x)\), is fundamental in hyperbolic functions. It is analogous to the cosine function in trigonometry, but tailored for hyperbolic geometry. The formula for the hyperbolic cosine is:\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. Unlike trigonometric cosine, which oscillates between -1 and 1, \(\cosh(x)\) always produces non-negative results, reflecting its smooth curve akin to an upward-opening parabola.

• \(\cosh(x)\) grows exponentially as \(x\) becomes large.
• It is an even function, meaning \(\cosh(x) = \cosh(-x)\).

Understanding this concept will help solve many problems involving hyperbolic functions.

Role of Exponential Functions

Exponential functions play a crucial role in mathematics, particularly when dealing with hyperbolic functions. An exponential function is typically expressed as \(e^x\), where \(e\) is Euler’s number. This function is not only continuous and differentiable but also grows rapidly as \(x\) increases.The properties of exponential functions are essential:

• \(e^{-x} = \frac{1}{e^x}\) exemplifies how exponential decay mirrors exponential growth.
• \(e^{\ln a} = a\) — this property links exponentials to logarithms directly.
• The function \(e^0 = 1\) shows that any number to the power of 0 is 1.

These properties are integral when evaluating the hyperbolic cosine in expression forms like \(\cosh(-\ln 3)\), where manipulating exponents is necessary.

Exploring the Properties of Logarithms

Understanding logarithms is key to manipulating expressions involving exponential and hyperbolic functions. A logarithm essentially answers the question: "To what power must the base \(e\) be raised to produce a given number?" The natural logarithm, denoted \(\ln(x)\), uses \(e\) as its base.Important properties of logarithms include:

• \(\ln(ab) = \ln(a) + \ln(b)\) helps in breaking products into sums.
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) simplifies divisions into differences.
• \(\ln(a^b) = b \cdot \ln(a)\) simplifies macroscopic exponents into products.
• Inverse property: \(e^{\ln a} = a\) and \(\ln(e^x) = x\), revealing the inverse relationship between exponentials and logarithms.

These properties allow for efficient simplification when dealing with expressions like \(\cosh(-\ln 3)\), creating easier pathways to finding solutions.