Problem 52 - Essential Calculus Early Transcendentals

Question

If \(x=\ln (\sec \theta+\tan \theta),\) show that \(\sec \theta=\cosh x\)

Step-by-Step Solution

Answer

\(\sec \theta = \cosh x\) with given \(x = \ln (\sec \theta + \tan \theta)\).

Step 1: Understand the Given Expression

We are given that \(x = \ln (\sec \theta + \tan \theta)\). Our goal is to show that \(\sec \theta = \cosh x\).

Step 2: Exponentiate Both Sides

Since \(x = \ln (\sec \theta + \tan \theta)\), we exponentiate both sides to remove the natural logarithm. Therefore, we have \(e^x = \sec \theta + \tan \theta\).

Step 3: Express \(\tan \theta\) in terms of \(\sec \theta\)

We know that \(\tan \theta = \sqrt{\sec^2 \theta - 1}\). Substitute this expression for \(\tan \theta\) in \(e^x = \sec \theta + \tan \theta\):\[e^x = \sec \theta + \sqrt{\sec^2 \theta - 1}\]

Step 4: Solve for \(\sec \theta\)

Isolate \(\sec \theta\) in terms of \(e^x\):\[\sec \theta = \frac{e^x + (e^{-x})}{2}\]This is obtained by using the identity for hyperbolic cosine: \(\cosh x = \frac{e^x + e^{-x}}{2}\). Thus, \(\sec \theta = \cosh x\).

Step 5: Verify the Solution

Apply the identity \(\cosh x = \frac{e^x + e^{-x}}{2}\) and check that both sides agree. As already shown, \(\sec \theta = \cosh x\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us solve equations where the unknown variable is an exponent. In our exercise, we have the natural logarithm function, denoted by \( \ln \). The natural logarithm uses \( e \) (approximately 2.718) as its base. It is particularly useful in calculus and complex calculations involving growth rates and time.

• Inverse Relationship: If \( y = \ln(a) \), it means \( e^y = a \). This relationship is crucial when we wish to eliminate the logarithm, as seen in Step 2 of the solution.
• Simplification: Since \( x = \ln(\sec \theta + \tan \theta) \), exponentiating both sides with \( e \) helps us simplify the expression to obtain \( e^x = \sec \theta + \tan \theta \). This makes manipulation easier when solving equations.

Using logarithmic identities appropriately can significantly streamline problem-solving, especially in cases that involve converting complex functions into simpler forms.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions which are true for every value of the occurring angles. These identities help us express functions in various ways, unlocking multiple paths to solving problems related to triangles and oscillations.

• Secant and Tangent Relationship: We use the identity \( \tan \theta = \sqrt{\sec^2 \theta - 1} \). This equation allows us to express tangent in terms of secant. It taps into the fundamental identity \( \sec^2 \theta = 1 + \tan^2 \theta \).
• Hyperbolic Conversion: By linking \( \sec \theta + \tan \theta \) with hyperbolic functions, we show that \( \sec \theta = \cosh x \). This conversion uses the resemblance between trigonometric and hyperbolic identities, like \( \cosh x = \frac{e^x + e^{-x}}{2} \).

Mastering these identities allows us to understand how angles and circular quadrants relate both in trigonometry and hyperbolic geometry.

Exponential Functions

Exponential functions involve expressions where variables are in the exponent. In our exercise, exponential functions, represented by \( e^x \), play a pivotal role in solving equations derived from logarithmic transformations.

• Exponentiation: Exponentiating the logarithmic function is a required step to simplify \( x = \ln(\sec \theta + \tan \theta) \). It transforms it into \( e^x = \sec \theta + \tan \theta \), which then can be analyzed further.
• Hyperbolic Cosine: The expression \( \frac{e^x + e^{-x}}{2} \) is derived from how exponential functions model hyperbolic cosine. By resolving secant in terms of hyperbolic functions, we demonstrate its link, illustrating how \( \sec \theta = \cosh x \).

Understanding exponential functions and their growth trends equips students with tools for analyzing real-world problems in fields like physics, finance, and natural phenomena.