Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-42: Use natural logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{e^{x}-e^{-x}}{2} $$

Step-by-Step Solution

Answer

\( x = \ln(y + \sqrt{y^2 + 1}) \)

Step 1: Multiply Both Sides by 2

We start by multiplying both sides of the equation by 2 to eliminate the fraction. This gives us:\[ 2y = e^{x} - e^{-x} \]

Step 2: Express in Terms of Hyperbolic Sine

Recall that the hyperbolic sine function is defined as \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\). Therefore, we can rewrite the equation as:\[ 2y = 2\sinh(x) \]And simplify it to:\[ y = \sinh(x) \]

Step 3: Solve for x using Inverse Hyperbolic Function

To isolate \(x\), use the inverse hyperbolic sine function, \(\text{sinh}^{-1}\). Thus, we take the inverse hyperbolic sine of both sides:\[ x = \text{sinh}^{-1}(y) \]

Step 4: Use Natural Logarithms

The inverse hyperbolic sine function can be expressed in terms of natural logarithms:\[ x = \ln(y + \sqrt{y^2 + 1}) \] This gives us the expression for \(x\) in terms of \(y\).

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are mathematical functions that are analogous to trigonometric functions but are based on hyperbolas instead of circles. They include functions like hyperbolic sine (\(\sinh(x)\)), hyperbolic cosine (\(\cosh(x)\)), and hyperbolic tangent (\(\tanh(x)\)). These functions are useful in various areas of mathematics, including solving differential equations, physics, and engineering.Hyperbolic sine, for example, is defined as:\[\sinh(x) = \frac{e^{x} - e^{-x}}{2}\]Where:

• \(e^{x}\) and \(e^{-x}\) are the exponential functions.
• This combines exponential growth and decay in one single function, providing a smooth curve.

Like their counterparts, trigonometric functions, hyperbolic functions have specific properties and identities that simplify complex expressions. When you use hyperbolic functions in mathematical expressions or problems, it often involves exponential functions, just like in our exercise where \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\) was used. This substitution helps simplify expressions and find solutions in other mathematical contexts.

Inverse Hyperbolic Functions Demystified

Inverse hyperbolic functions are used to find the angle or value associated with a given hyperbolic function. Inverses have prefixes such as 'arsinh' or 'asinh' for inverse hyperbolic sine. The inverse hyperbolic sine, for example, is denoted as \(\sinh^{-1}(x)\). It helps you find the corresponding values of \(x\) if you have the value of \(\sinh(x)\). This concept is analogous to using \(\sin^{-1}(x)\) in trigonometry. For the hyperbolic sine function, the inverse can be expressed as:\[\sinh^{-1}(y) = \ln(y + \sqrt{y^2 + 1})\]Where:

• \(\ln\) represents the natural logarithm.
• The expression \(y + \sqrt{y^2 + 1}\) ensures the argument of the logarithm is always positive, which is required in logarithmic functions.

These inverse functions are crucial in untangling the expressions involving hyperbolic functions, helping us solve equations where these functions occur. They transform hyperbolic computations into manageable algebraic tasks, which makes them incredibly useful in mathematical problem-solving like our exercise.

Algebraic Manipulation Techniques

Algebraic manipulation refers to the various techniques used to simplify expressions, solve equations, and transform formulas into more convenient forms. This includes operations like factoring, expanding, and moving terms around an equation, as seen in our original solution.In our exercise:

• We began by multiplying both sides of the equation by 2 to eliminate fractions, transforming \( y = \frac{e^{x} - e^{-x}}{2} \) into \( 2y = e^{x} - e^{-x} \).
• This step simplified the process of expressing the equation in terms of hyperbolic functions.

Simplification serves a fundamental role in solving equations, whether they are algebraic, trigonometric, or involve hyperbolic functions. By applying these techniques, you can streamline complex equations into simpler, more solvable expressions. Understanding how and when to deploy these tactics is key in tackling a wide array of mathematical challenges.

Applying Trigonometric Identities in Calculations

Trigonometric identities are equations involving trigonometric functions that are universally true for all variable values. These identities allow for the transformation of more complex trigonometric expressions into simpler forms.While hyperbolic functions have their own identities, understanding traditional trigonometric identities can set a solid foundation. Familiar identities include:

• Pythagorean identities, like \(\sin^2(x) + \cos^2(x) = 1\).
• Angle sum and difference identities, which are essential for expanding and simplifying expressions.

In the context of hyperbolic functions, similar identities and transformations exist but involve hyperbolic sine and cosine. Hence, you often see hyperbolic expressions remapped through trigonometric-like reasoning to solve for variables.When dealing particularly with the exercise mentioned, we utilize such identities and analogies to change an equation and eventually solve for \(x\). They are intrinsic to unraveling complex expressions and finding solutions in both trigonometric and hyperbolic realms.