Problem 35 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. If \(a_{1}+a_{2} \sin x+a_{3} \cos x+a_{4} \sin 2 x+a_{5} \cos 2 x=0\) holds for all \(x\), then the number of possible 5 -tuplets is (a) 0 (b) 1 (c) 2 (d) infinity

Step-by-Step Solution

Answer

(b) 1

Step 1: Analyse the Parameters

The parameters \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) are the coefficients of the trigonometric functions in the equation. If the equation holds for all \(x\), these coefficients must equal zero such that the sum of all terms always results in zero.

Step 2: Apply the Zero Coefficient Rule

As seen in step 1, for the equation to hold for all x values, each coefficient must equal zero, meaning that \(a_{1} = a_{2} = a_{3} = a_{4} = a_{5} = 0\). This gives us a 5-tuplet of zeros, confirming that all coefficients must be zero.

Step 3: Count the Number of 5-tuplets

Only one 5-tuplet of coefficients, (0,0,0,0,0), makes the equation true for all values of x, hence the number of possible 5-tuplets is 1.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially useful in solving equations involving angles. In our exercise, the functions used are sine and cosine, specifically \( \sin x \), \( \cos x \), \( \sin 2x \), and \( \cos 2x \). These functions can represent various cyclical patterns.

• \( \sin x \) and \( \cos x \) allows us to represent periodic oscillations with respect to an angle (\( x \)). They have a range of -1 to 1.
• Multiplying by double angles, i.e., \( \sin 2x \) and \( \cos 2x \), shifts the frequency of the oscillation. This changes the equation's dynamics and allows us to explore different transformations of the original function.

These functions are featured with coefficients in the exercise, contributing to the equation's structure and directly affecting the solution outcome by affecting the output of these periodic functions.

Coefficient Analysis

In this exercise, coefficients \( a_{1}, a_{2}, a_{3}, a_{4}, \) and \( a_{5} \) play critical roles. They determine the amplitude of the corresponding trigonometric functions. Understanding them is pivotal in solving such trigonometric equations.

Every coefficient multiplies with its respective trigonometric function. For instance, \( a_{2} \) and \( a_{3} \) scale the functions \( \sin x \) and \( \cos x \), respectively. It indicates how much each function contributes to the overall equation.

• If all functions are to cancel out for all values of \( x \), their combined effect must always sum to zero.
• This requires a balanced set of coefficients where each trigonometric contribution is equaled out, an essential point for further sections.

Such analysis is crucial as it sets the precedent for understanding solutions where coefficients have to conform to specific values to satisfy the equation globally.

Zero Coefficient Rule

The Zero Coefficient Rule is an insightful shortcut in determining solutions to polynomial and trigonometric equations. If an equation has to be true for every possible value of \( x \), each term's influence needs to vanish independently.

In our exercise, this is achieved by making each coefficient zero. This results because all the component sine and cosine functions must not affect the result for it to consistently remain zero across all \( x \) values.

• Therefore, \( a_{1} = a_{2} = a_{3} = a_{4} = a_{5} = 0 \) becomes the only 5-tuplet that maintains the equation zero naturally for all angle values.
• Only one such configuration exists, underscoring the uniqueness of zero in this broad satisfying condition.

The rule, thus, simplifies the analysis and solving of complex equations, efficiently finding definitive constraints that need tight adherence.