Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

The answer is (B) \(a=24\).

Step 1: Rewrite the Expression

Start by examining the given expression: \( a \cos^2 3\alpha + b \cos^4 \alpha = 16 \cos^6 \alpha + 9 \cos^2 \alpha \). This is supposed to hold for any \(\alpha\), implying an identity. Consequently, each component of the equation involving \(\cos\alpha\) must be equivalent separately.

Step 2: Simplify Using Identities

Observe that \(\cos^2 3\alpha = (4\cos^3 \alpha - 3\cos \alpha)^2\). Let's express this using the identity for \(\cos 3\alpha\). Substitute to get terms involving \(\cos^2 \alpha\) so that the left-hand side involves \(\cos^2 \alpha\) and its higher powers.

Step 3: Equate Coefficients of Like Terms

After substitution, equate the coefficients of like terms from both sides of the equation to establish relationships between \(a\) and \(b\). Specifically, we equate the coefficients of \(\cos^6\alpha\), \(\cos^4\alpha\), and \(\cos^2\alpha\) terms.

Step 4: Solve for Constants

Solve the equations obtained from step 3 to find the values of \(a\) and \(b\). This involves matching coefficients from the expanded forms, ultimately finding \(a\ = 24 \) as consistent with all terms while checking validity for \(b\).

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions and are set equal to some value or another expression. In solving these types of equations, the goal is to find all angle values that make the equation true. Commonly, trigonometric identities are used to transform and simplify these equations.

• Standard Forms: Trigonometric equations can be represented in various forms, such as sums, products, or single trigonometric functions.
• Identity Application: To solve equations that are identities (i.e., they hold true for any angle), it involves verifying equivalences rather than finding specific solutions.

In our exercise, we deal with a proposed identity involving trigonometric functions and constants, where we must confirm its validity under all circumstances. This generally involves ensuring that each power of the cosine function matches on both sides of the equation.

Coefficient Comparison

Coefficient comparison is a critical technique when dealing with equations that are identically true. This method focuses on comparing like terms—terms that involve the same power of a variable or trigonometric function.

• Equate Like Terms: In an identity, every term involving the same power of a trigonometric function must have equal coefficients on both sides of the equation.
• Logical Steps: By expanding expressions like \( \cos^2 3\alpha \) using identities, each coefficient can be isolated and set to equal its counterpart on the opposite side of the equation.

In this exercise, coefficient comparison allows us to determine the unknown constants \(a\) and \(b\) by ensuring that terms involving \( \cos^6 \alpha \), \( \cos^4 \alpha \), and \( \cos^2 \alpha \) across the entire equation align perfectly.

Cosine Function

The cosine function, represented as \(\cos\) in trigonometry, is fundamental in describing relationships in triangles and oscillatory motion. It varies between -1 and 1, cycling over its period of \(2\pi\) radians.

• Multiple Angles: Functions like \(\cos 3\alpha\) are derived from the standard cosine through angle-multiplicative identities, which help simplify and express complex trigonometric equations.
• Power Functions: Higher powers of cosine, such as \(\cos^2\alpha\), change the equation's dynamics by contributing different terms through basic algebraic manipulations.

Understanding these changes and how to express them using identities is essential for solving and simplifying intricate trigonometric problems, like the one in our exercise, where transformations of \(\cos\alpha\) underpin the problem's solution.