Problem 36 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a \cos ^{2} 3 \alpha+b \cos ^{4} \alpha=16 \cos ^{6} \alpha+9 \cos ^{2} \alpha\) is iden- tity, then (A) \(a=1\) (B) \(a=24\) (C) \(b=1\) (D) \(b=24\)

Step-by-Step Solution

Answer

(B) \(a=1\), (D) \(b=24\)

Step 1: Analyze the Given Identity

We are given the identity \(a \cos^2 3\alpha + b \cos^4 \alpha = 16 \cos^6 \alpha + 9 \cos^2 \alpha\). We need to find the values for \(a\) and \(b\) that make this an identity, meaning it holds true for all values of \(\alpha\).

Step 2: Expand \(\cos^2 3\alpha\) Using Trigonometric Identities

Recall that \(\cos 3\alpha = 4\cos^3 \alpha - 3\cos \alpha\). Therefore, \(\cos^2 3\alpha = (4\cos^3 \alpha - 3\cos \alpha)^2\).

Step 3: Expand the Expression for \(\cos^2 3\alpha\)

Calculate \((4\cos^3 \alpha - 3\cos \alpha)^2\):\[(4\cos^3 \alpha - 3\cos \alpha)^2 = 16\cos^6 \alpha - 2 \times 12 \cos^4 \alpha + 9\cos^2 \alpha\] \[= 16 \cos^6 \alpha - 24 \cos^4 \alpha + 9 \cos^2 \alpha\].

Step 4: Substitute and Compare Coefficients

Substitute \(16 \cos^6 \alpha - 24 \cos^4 \alpha + 9 \cos^2 \alpha\) back into the equation:\[a(16 \cos^6 \alpha - 24 \cos^4 \alpha + 9 \cos^2 \alpha) + b \cos^4 \alpha = 16 \cos^6 \alpha + 9 \cos^2 \alpha\]. Simplify to get:\[16a \cos^6 \alpha - 24a \cos^4 \alpha + 9a \cos^2 \alpha + b \cos^4 \alpha = 16 \cos^6 \alpha + 9 \cos^2 \alpha\].

Step 5: Solve for Coefficients

Equating the coefficients from both sides of the equation, we get:For \(\cos^6 \alpha\), \(16a = 16\) \(\Rightarrow a = 1\).For \(\cos^4 \alpha\), \(-24a + b = 0\) \(\Rightarrow -24 + b = 0\) \(\Rightarrow b = 24\).

Step 6: Verify the Identity

Substitute \(a = 1\) and \(b = 24\) back into the original identity:\[(1) \cdot (\cos^2 3\alpha) + (24) \cdot (\cos^4 \alpha) = 16 \cos^6 \alpha + 9 \cos^2 \alpha\].The identity holds, confirming our solutions.

Key Concepts

Mathematical Problem Solving

Mathematical problem solving is a vital skill that involves understanding, interpreting, and analyzing mathematical situations. In our exercise, the challenge is to confirm that a given trigonometric equation is an identity and to find the correct values of specific constants that hold true for all scenarios.

This kind of problem requires a structured approach. First, we determine what we know and what we need to discover. Here, the task is to establish the values of constants \(a\) and \(b\) in the equation. This involves expanding the expression using trigonometric identities, substituting the variables properly, and then comparing coefficients.

By thinking logically and following a series of straightforward steps, such as expanding terms and equating coefficients, mathematical problem solving becomes manageable. Always take each stage step-by-step to avoid unnecessary complications, and remember to stay organized as you work through the solution.

Trigonometrical Coefficients Comparison

Trigonometrical coefficients comparison plays a crucial role in solving trigonometric identities. **The core idea is to ensure that the expression holds true by balancing the coefficients of like terms on both sides of the equation.**

Let's break it down further:

• Begin with expanding the trigonometric terms using relevant identities. As seen in our problem, expanding \( \cos^2 3\alpha \) requires using the identity \( \cos 3\alpha = 4\cos^3 \alpha - 3\cos \alpha \).
• After expansion, substitute these results back into the original equation.
• Next, reorganize the equation by grouping like terms, such as \( \cos^6 \alpha \), \( \cos^4 \alpha \), and \( \cos^2 \alpha \).
• Finally, compare the coefficients of these like terms from both sides. This step involves solving simple algebraic equations to find the unknown coefficients.

This method ensures that each term and its corresponding coefficient are balanced correctly, confirming the identity. When done with care, it leaves no room for errors.

JEE Mathematics

JEE Mathematics is designed to test a student's understanding, application, and problem-solving abilities in mathematics. **One essential topic in JEE Mathematics is trigonometry, particularly identities and equations.**

Understanding trigonometric identities is critical, as they offer tools for simplifying expressions and solving complex problems. In a JEE context, candidates are often required to demonstrate their knowledge of identities and their ability to apply these in different situations, such as verifying identities or finding values of unknown variables.

Mastering these skills requires practice and familiarity with a wide range of problems. To prepare effectively for JEE Mathematics, students should:

• Practice various types of trigonometric problems.
• Understand and remember the core trigonometric identities.
• Work on enhancing algebraic manipulation skills.
• Solve past years' question papers to get a feel of the patterns.

With patience and diligent practice, students can excel in tackling these mathematical challenges effectively, paving the way for success in the examination.