Problem 3 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x \cos ^{2} 3 \theta+y \cos ^{4} \theta=16 \cos ^{6} \theta+9 \cos ^{2} \theta\) be an iden- tity, then (A) \(x=-1, y=24\) (B) \(x=1, y=24\) (C) \(x=24, y=1\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (B) \(x=1, y=24\).

Step 1: Identify the Given Equation

We are given the equation \(x \cos^2(3\theta) + y \cos^4(\theta) = 16 \cos^6(\theta) + 9 \cos^2(\theta)\). We need to identify possible values of \(x\) and \(y\) such that this equation holds as an identity.

Step 2: Simplify the Equation

First, express \(\cos^2(3\theta)\) in terms of \(\cos^2(\theta)\). Using the triple angle formula, \(\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)\), we get \(\cos^2(3\theta) = (4\cos^3(\theta) - 3\cos(\theta))^2\). Simplifying gives: \[\cos^2(3\theta) = 16\cos^6(\theta) - 24\cos^4(\theta) + 9\cos^2(\theta).\]

Step 3: Substitute and Rearrange

Substitute \(\cos^2(3\theta)\) from Step 2 into the equation: \[x(16\cos^6(\theta) - 24\cos^4(\theta) + 9\cos^2(\theta)) + y\cos^4(\theta) = 16\cos^6(\theta) + 9\cos^2(\theta).\] Next, expand and group similar terms, based on powers of \(\cos(\theta)\).

Step 4: Compare Coefficients

Equate coefficients of \(\cos^{6}(\theta)\), \(\cos^{4}(\theta)\), and \(\cos^{2}(\theta)\):- For \(\cos^6(\theta)\): \(16x = 16\) implies \(x = 1\).- For \(\cos^4(\theta)\): \(-24x + y = 0\).- For \(\cos^2(\theta)\): \(9x = 9\) also confirms \(x = 1\).Now solve \(-24x + y = 0\) for \(x = 1\) to get \(-24(1) + y = 0\), so \(y = 24\).

Step 5: Verify the Solution

Verify the values \(x = 1\) and \(y = 24\) satisfy all terms:- Substitute back into the grouped equation: - For \(\cos^6(\theta)\): \(16x = 16\) holds. - For \(\cos^4(\theta)\): \(-24(1) + 24 = 0\) holds. - For \(\cos^2(\theta)\): \(9x = 9\) holds.Therefore, values \(x = 1\) and \(y = 24\) are correct and satisfy the identity.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is one of the primary functions used in trigonometry, alongside sine and tangent. Cosine is often represented as \( \cos(\theta) \), where \( \theta \) is the angle in question. This function is periodic and oscillates between -1 and 1.

Learning how cosine behaves, especially within different equations and transformations, is crucial. In the context of trigonometric identities, understanding how to manipulate the cosine function using identities, like the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \), is essential. This allows cosine terms to be simplified or expanded easily.

In the given equation, the cosine function appears in various powers, which indicates its transformation under specific identities, such as the triple angle formula utilized later in the exercise.

Triple Angle Formula

The triple angle formula is a specific type of identity used to express trigonometric functions of triple angles, such as \(3\theta\), in terms of single angles. For cosine, the formula is \( \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \). This identity is an extension of the double angle formulas and is particularly useful in simplifying expressions involving multiple angular terms.

Applying the triple angle formula allows us to transform \( \cos^2(3\theta) \) into a more workable form. In this exercise, by squaring the triple angle identity, you convert it into functions of \( \cos(\theta) \):

• First, express \( \cos(3\theta) \) as a polynomial in terms of \( \cos(\theta) \).
• Then, expand \( (4\cos^3(\theta) - 3\cos(\theta))^2 \) to simplify it into terms of \( \cos^6(\theta) \), \( \cos^4(\theta) \), and \( \cos^2(\theta) \).

This step is crucial for manipulating and simplifying trigonometric identities, as seen in the given exercise.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form. This often involves combining like terms, using algebraic identities, and applying trigonometric identities to rewrite the expression. Simplification not only makes the equation easier to handle but also prepares it for further analysis or solution.

In the textbook exercise, simplifying the left side of the equation using the triple angle identity converts the original expression into a polynomial of \( \cos(\theta) \). This allows us to line up coefficients with the right side of the equation for easier comparison.

When simplifying:

• Ensure each cosine term is expressed with the same powers for straightforward comparison.
• Group similar terms together, based on \( \cos^6(\theta) \), \( \cos^4(\theta) \), and \( \cos^2(\theta) \).
• Check each simplification step for errors, maintaining mathematical equality throughout.

This process is critical for resolving identities and ensuring each side of the equation is balanced.

Coefficient Comparison

Coefficient comparison is a method commonly used in mathematics to identify unknowns within polynomial or trigonometric identities. In context, this technique helps determine the unknown coefficients by equating powers of a variable or trigonometric function.

In this exercise, after simplifying both sides of the equation, comparing coefficients involves:

• Aligning corresponding terms of each power of \( \cos(\theta) \) on both sides of the equation.
• Setting coefficients from these terms equal to each other.
• Solving the resulting equations to find the unknowns \(x\) and \(y\).

This method is powerful as it directly links different powers of a variable to a set of linear equations. This is particularly useful when dealing with identities, where one needs to prove the algebraic equivalence of two expressions. Here, it confirms the harmony between the simplified terms by determining \(x = 1\) and \(y = 24\), meeting the criteria of the identity.