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

Question

The value of \(\theta\), lying between \(\theta=0\) and \(\theta=\frac{\pi}{2}\) and satisfying the equation \(\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0\), is (A) \(\frac{11 \pi}{24}\) (B) \(\frac{7 \pi}{24}\) (C) \(\frac{5 \pi}{24}\) (D) none of these

Step-by-Step Solution

Answer

The correct value of \( \theta \) is \( \frac{7\pi}{24} \).

Step 1: Understanding the Problem

We need to find the value of \( \theta \) satisfying a determinant equation, where \( \theta \) is within the interval \( [0, \frac{\pi}{2}] \). The equation involves trigonometric functions: sine and cosine.

Step 2: Formulate the Determinant

The given matrix is: \[\begin{bmatrix}1+\cos^{2}\theta & \sin^{2}\theta & 4\sin 4\theta \\cos^{2}\theta & 1+\sin^{2}\theta & 4\sin 4\theta \\cos^{2}\theta & \sin^{2}\theta & 1+4\sin 4\theta\end{bmatrix}\]Our goal is to calculate this matrix's determinant and set it to zero.

Step 3: Calculate the Determinant

Expand the 3x3 determinant using the standard formula for calculating determinants. This will involve taking the first row and applying the cofactor expansion method (also known as Laplace's expansion).

Step 4: Simplify Trigonometric Identities

Notice that \( \cos^2 \theta + \sin^2 \theta = 1 \) by the Pythagorean identity. This can help simplify the matrix entries, such as turning \( 1 + \cos^{2}\theta \) into \( 1 + (1 - \sin^{2}\theta) = 2 - \sin^{2} \theta \). Further simplifications yield easier expressions for the determinant.

Step 5: Set the Determinant to Zero

The simplification should result in a trinomial (polynomial) involving \( \sin 4\theta \) and other simplified terms. Set the simplified determinant expression equal to 0.

Step 6: Solve for \( \theta \)

Once the polynomial is set to zero, solve for \( \theta \) within the interval \( [0, \frac{\pi}{2}] \). Calculate the values based on given options and check which satisfies the determinant equation.

Step 7: Verify Solutions

Substitute the calculated \( \theta \) values back into the determinant expression to verify that the determinant evaluates to 0. This ensures the solution is valid.

Key Concepts

Determinants

A determinant is a special number that can be calculated from a square matrix. It's a key concept in mathematics, particularly in linear algebra. Determinants help us understand matrix properties such as invertibility, eigenvalues, and more.

• A matrix has a determinant only if it is square, meaning it has the same number of rows and columns.
• Determinants are used to solve systems of linear equations, perform coordinate transformations, and even in calculus.

When finding a determinant, we use a process called cofactor expansion (or Laplace's expansion), typically along the first row or column. This involves decomposing the matrix into smaller matrices and calculating the product of their determinants.
In the context of trigonometric equations, determining the value of a determinant can often involve the use of identities such as the Pythagorean identity. The determinant's value can provide significant insight into the properties of the trigonometric function within a certain interval, such as \(0, \pi/2\).

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It provides a simple yet powerful way to relate the sine and cosine of an angle.The identity is given by:\[\cos^2 \theta + \sin^2 \theta = 1\]This equation holds for any angle \( \theta \). It's derived from the Pythagorean theorem and plays a crucial role in simplifying many trigonometric expressions.

• The identity is often used to transform expressions, especially in the calculation of determinants.
• In matrices involving trigonometric functions, using the Pythagorean identity can simplify complex expressions into something easier to work with.

In the given exercise, this identity helps simplify terms like \(1 + \cos^2 \theta\) into \(2 - \sin^2 \theta\). Recognizing and applying this identity can significantly reduce the computational complexity of the problem.

Matrix Determinant

The determinant of a matrix, especially in higher dimensions, reveals important characteristics about the matrix itself. In a 3x3 matrix, like the one in the exercise, the determinant can be calculated using the cofactor expansion method. This method involves expanding the matrix along a row or column and calculating minor determinants, which are smaller matrices formed by removing one row and one column.

• For a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is given by:

\[\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)\]In trigonometric contexts, the entries of the matrix might involve functions like \( \sin 4\theta \) and need careful computation. Finding the determinant and setting it equal to zero can allow you to solve for specific values of \( \theta \) that meet the given conditions, especially within a specified interval. Mastering these calculations is key to understanding the problem's trigonometric constraints and solutions.