Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The general solution of the equation \(2^{\cos ^{2} \theta}+1=3.2^{-\sin ^{2} \theta}\) is (A) \(2 n \pi \pm \frac{\pi}{2}, n \pi, n \in Z\) (B) \(n \pi \pm \frac{\pi}{2}, 2 n \pi, n \in Z\) (C) \(n \pi \pm \frac{\pi}{2}, n \pi, n \in Z\) (D) none of these

Step-by-Step Solution

Answer

The equation solution matches part of option (A).

Step 1: Rewrite the Equation Using Trigonometric Identity

Start by using the identity \( \cos^{2} \theta + \sin^{2} \theta = 1 \). This allows us to express either \( \cos^{2} \theta \) or \( \sin^{2} \theta \) in terms of the other. Substituting for \( \cos^{2} \theta \), the equation becomes \( 2^{1 - \sin^{2} \theta} + 1 = 3 \cdot 2^{-\sin^{2} \theta} \).

Step 2: Simplify the Equation

Rewrite the left side of the equation as a single power of 2: \( 2^{1 - \sin^{2} \theta} = \frac{2}{2^{\sin^{2}\theta}} \). So the equation becomes \( \frac{2}{2^{\sin^{2}\theta}} + 1 = 3 \cdot 2^{-\sin^{2}\theta} \).

Step 3: Clear the Denominators

Multiply the entire equation by \( 2^{\sin^{2}\theta} \) to eliminate the denominator: \( 2 + 2^{\sin^{2}\theta} = 3 \).

Step 4: Solve for \( 2^{\sin^{2}\theta} \)

Subtract 2 from both sides: \( 2^{\sin^{2}\theta} = 1 \). Recognize that \( 2^{\sin^{2}\theta} = 1 \) holds when \( \sin^{2}\theta = 0 \), meaning \( \sin \theta = 0 \).

Step 5: Determine the General Solution for \( \sin \theta = 0 \)

The solutions for \( \sin \theta = 0 \) occur at \( \theta = n \pi \), where \( n \) is any integer in \( \mathbb{Z} \).

Step 6: Compare Solutions to Given Options

Compare \( \theta = n \pi \) with the given options. It is directly included only in option (A) as part of the broader solution that includes \( \theta = 2n\pi \pm \frac{\pi}{2} \). None of the options solely provide \( \theta = n \pi \). Thus, the correct solution is embedded within option (A), but not as a standalone solution description.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools used in mathematics to simplify and solve equations involving trigonometric functions like sine and cosine. One of the most fundamental trigonometric identities is \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity states that the square of the cosine of an angle plus the square of the sine of the same angle is always equal to one. This relationship is key in transforming trigonometric expressions into more manageable forms.
For example, if you need to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \), you can rearrange the identity to \( \cos^2 \theta = 1 - \sin^2 \theta \). This transformation is crucial for substituting values and simplifying equations, as seen in the original problem where the identity was used to rewrite the equation in terms of \( \sin^2 \theta \).

• Allows interchanging variables between \( \sin \) and \( \cos \)
• Helps simplify complex trigonometric equations
• Fundamental in proofs and derivations involving trigonometric functions

It's important to get comfortable using these identities, as they are invaluable when solving more advanced trigonometric problems.

General Solution

In trigonometric equations, the general solution represents all possible angles that satisfy the equation. Since trigonometric functions like sine and cosine are periodic, meaning they repeat their values over regular intervals, the solutions to trigonometric equations are often expressed as a set of infinitely many angles.
For a simple case like \( \sin \theta = 0 \), the solutions occur at \( \theta = n\pi \), where \( n \) is an integer. This specific form accounts for every instance where the sine function yields 0 along its periodic wave. The term "general solution" ensures that no solution is overlooked, covering all the repetitions of an angle that satisfy the given equation.

• Covers infinite solutions for periodic functions
• Uses the general form \( n \cdot T + P \) where \( T \) is the period
• Incorporates all repeating patterns of solutions

Understanding general solutions is essential because trigonometric equations are inherently periodic, and only providing a limited set of solutions could be misleading.

Sin and Cos Functions

The sine and cosine functions are fundamental trigonometric functions that describe relationships within right-angled triangles and the unit circle.
The sine function, \( \sin \theta \), measures the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right-angled triangle, while the cosine function, \( \cos \theta \), measures the ratio of the length of the adjacent side to the hypotenuse.
These functions are periodic, with the sine function having a period of \( 2\pi \) and the cosine function also having a period of \( 2\pi \). This means that for every full circle rotation (\( 360^\circ \) or \( 2\pi \) radians), their values repeat.

• Sine function peaks at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \)
• Cosine function peaks at \( 0 \) and \( \pi \)
• They serve as the basis for understanding waves and oscillations

Grasping how these functions work and their periodic nature is crucial for tackling both pure and applied mathematical problems. They are essential in fields ranging from engineering to physics for analyzing oscillatory systems.