Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x \cos ^{2} 3 \theta+y \cos ^{4} \theta=16 \cos ^{6} \theta+9 \cos ^{2} \theta\) be an identity, 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 answer is (B) \(x=1, y=24\).

Step 1: Express in Known Terms

We start with the given identity: \(x \cos^{2} 3\theta + y \cos^{4}\theta = 16 \cos^{6} \theta + 9 \cos^{2}\theta\). We need to express \( \cos^{2} 3\theta \) in terms of \( \cos^{4} \theta \) and \( \cos^{2} \theta \). To do this, let's use the identity \( \cos 3\theta = 4\cos^3\theta - 3\cos\theta \), and substitute \( \cos^{2} 3\theta = (4\cos^3\theta - 3\cos\theta)^2 \).

Step 2: Expand \(\cos^{2} 3\theta\)

Expand \((4\cos^3\theta - 3\cos\theta)^2\): \[(4\cos^3\theta - 3\cos\theta)^2 = (16\cos^6\theta - 24\cos^4\theta + 9\cos^2\theta)\].This is the expression for \(\cos^{2} 3\theta\) in terms of powers of \(\cos \theta\).

Step 3: Substitute in the Identity

Substitute \(\cos^{2} 3\theta\) into the identity:\[ x(16\cos^6\theta - 24\cos^4\theta + 9\cos^2\theta) + y\cos^4\theta = 16\cos^6\theta + 9\cos^2\theta.\]

Step 4: Match Coefficients

Now equate the coefficients of like terms to form a system of equations:- For \(\cos^6\theta\): \(16x = 16\), giving \(x = 1\).- For \(\cos^4\theta\): \(-24x + y = 0\).- For \(\cos^2\theta\): \(9x = 9\), which is consistent with \(x = 1\).

Step 5: Solve for \(y\)

Using the equation from \(\cos^4\theta\): \(-24 \times 1 + y = 0\), we solve for \(y\): \[y = 24\].

Key Concepts

Cosine Function

The cosine function, denoted \( \cos \), is a fundamental trigonometric function that arises frequently in mathematics, especially in problems involving triangles and periodic phenomena. It represents the adjacent side over the hypotenuse in a right-angled triangle in a unit circle context. This function is significant in understanding wave patterns, oscillations, and rotations.
Cosine is defined for an angle \( \theta \) as \( \cos \theta \). When dealing with expressions like \( \cos^{2} 3\theta \), you are essentially dealing with the square of the cosine of a multiple angle.
To simplify such a term, trigonometric identities come in handy. For example, using the identity \( \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \), you can express \( \cos^2 3\theta \) in terms of \( \cos \theta \). This is central to problems involving polynomial-like expressions with trigonometric functions, where simplifying expressions allows solving systems of equations effectively.

System of Equations

Systems of equations are collections of two or more equations with the same set of variables. Solving such systems is crucial, especially when determining specific values (like trigonometric coefficients) that satisfy all equations simultaneously.
In the given problem, once the trigonometric terms have been simplified and equated, you derive a system of equations by comparing the coefficients of like powers of \( \cos \theta \). This process is essential for finding unknown variables like \( x \) and \( y \).
The equations obtained from matching coefficients in our exercise form such a system. Solving this system involves:

• Setting each coefficient on the left equal to the corresponding coefficient on the right.
• Solving for each variable step-by-step, considering one equation may yield a variable that is then substituted into another to find others.

Understanding this process helps in tackling more complex algebraic situations where multiple unknowns must be identified.

Coefficient Matching

Coefficient matching is a technique where we equate coefficients of identical powers of variables from two expressions to form equations. This method is vital when working with identities where expanded forms involve the same variable in different powers.
In trigonometric identities, once you have rewritten terms in a polynomial-like structure (e.g., terms involving \( \cos \theta, \cos^2 \theta, \) etc.), you compare each power's coefficient on both sides of the equation. This results in a straightforward system of equations to solve.
In the exercise, by equating the coefficients of \( \cos^6 \theta, \cos^4 \theta, \) and \( \cos^2 \theta \), explicit equations are formed, like \( 16x = 16 \). By resolving these equations, you determine unknown variables such as \( x = 1 \) and \( y = 24 \).
Applying coefficient matching not only simplifies these problems but also enhances algebraic insight, helping to unravel and solve more intricate equations efficiently.