Problem 46 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation \(e^{\sin x}-e^{-\sin x}-4=0\) has (A) infinite number of real roots (B) no real roots (C) exactly one real root (D) exactly four real roots

Step-by-Step Solution

Answer

The equation has no real roots.

Step 1: Simplify the Equation

First, rewrite the given equation \( e^{\sin x} - e^{-\sin x} - 4 = 0 \) using the concept of hyperbolic sine: \( e^{\sin x} - e^{-\sin x} = 2\sinh(\sin x) \). This transforms the equation to \( 2\sinh(\sin x) = 4 \). Simplifying further gives \( \sinh(\sin x) = 2 \).

Step 2: Use Definition of Hyperbolic Sine

Recall that the hyperbolic sine function is defined by \( \sinh y = \frac{e^y - e^{-y}}{2} \). For \( \sinh y = 2 \), we have \( y = \sinh^{-1}(2) \) where \( y = \sin x \). Thus, we need to solve \( \sin x = \sinh^{-1}(2) \).

Step 3: Solve for Values of x

The expression \( \sin x = \sinh^{-1}(2) \) suggests finding values of \( x \) such that \( \sin x \) equals the hyperbolic sine inverse of 2. Since the sine function \( \sin x \) only ranges from -1 to 1, and \( \sinh^{-1}(2) > 1 \), \( \sin x = \sinh^{-1}(2) \) is impossible for any real \( x \).

Step 4: Conclusion on Number of Real Roots

Since the range of \( \sin x \) (-1 to 1) does not include vales greater than 1, and given \( \sinh^{-1}(2) > 1 \), the equation \( \sin x = \sinh^{-1}(2) \) has no real solution. Therefore, the original equation \( e^{\sin x} - e^{-\sin x} - 4 = 0 \) has no real roots.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas instead of circles. They are important in many fields, including calculus, physics, and engineering. One of the key hyperbolic functions is the hyperbolic sine, denoted as \(\sinh\). It is defined as \(\sinh y = \frac{e^y - e^{-y}}{2}\). This function shares properties similar to the regular sine function but extends beyond typical trigonometric constraints.
Hyperbolic functions are invaluable in solving equations where the classic trigonometric identities might not suffice. They provide insight into problems involving exponential growth and decay, contributing to a deeper mathematical understanding.

• Hyperbolic sine and cosine are the most commonly used hyperbolic functions.
• These functions can describe the shapes of hanging cables or chains (catenary curves).

Hyperbolic functions have inverse functions that help solve complex equations, as demonstrated in the problem where \(\sinh(\sin x) = 2\).

Inverse Functions

Inverse functions reverse the operation of another function. If you have a function \(f(x)\), its inverse \(f^{-1}(y)\) gives you the original input \(x\) when given the output \(y\). They are crucial for finding values that lead to specific results in equations. For example, the inverse of the hyperbolic sine function, \(\sinh^{-1}(x)\), lets us solve for \(x\) when given a specific hyperbolic sine value.
Inverse functions are helpful for undoing operations or finding original inputs in various mathematical problems. They provide a straightforward method to backtrack through calculations or to verify solutions.

• Inverse functions are denoted by a superscript \(-1\), such as \(\sin^{-1}(x)\) or \(\cosh^{-1}(x)\).
• The inverse function can be found graphically by reflecting the original function over the line \(y = x\).

The difficulty of finding inverse values often lies in ensuring that the range and domain are correct for the function in question.

Range of Sine Function

The range of the sine function is a crucial concept in trigonometry. The sine function, \(\sin x\), oscillates between -1 and 1 for all real numbers \(x\). This means that no matter what angle you input, the sine's output will never exceed these bounds.
Understanding the range is essential, especially when solving equations involving the sine function, as it restricts possible real solutions. As in the problem that we examined, realizing that \(\sinh^{-1}(2)\) exceeds the range of \(\sin x\) was the key to determining no real solutions.

• The sine function is periodic with a period of \(2\pi\); it repeats values after every \(2\pi\) interval.
• In many trigonometric problems, knowing the exact range helps to identify restrictions or so-called 'impossible' values.

Constraints like these carve the boundaries of what's possible when working with trigonometric equations and ensure solutions remain within real-world limitations.