Problem 10 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\left|\begin{array}{lcc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|\) (A) 0 (B) \(\sin \theta\) (C) \(\cos \theta\) (D) independent of \(\theta\)

Step-by-Step Solution

Answer

The value of the determinant is 0 (Option A).

Step 1: Understanding the Determinant Structure

We start by identifying the determinant of the given matrix. The matrix is a 3x3 matrix composed of trigonometric functions based on the variable \( \theta \). Our task is to evaluate this determinant.

Step 2: Applying Sine and Cosine Angle Sum Formulas

Consider the entries of the matrix: \( \sin(\theta + \frac{2\pi}{3}) = \sin \theta \cdot \cos \frac{2\pi}{3} + \cos \theta \cdot \sin \frac{2\pi}{3} \), similarly for the \( \cos \) and \( \sin \) for shifted angles due to symmetry. Recall that \( \cos \frac{2\pi}{3} = -\frac{1}{2}, \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \). Apply similar transformations for other shifts.

Step 3: Using Trigonometric Identities

Notice that using angle identity transformations in the element \( \sin(2\theta + \frac{4\pi}{3}) \) and similar, symmetry among the rows appears. Use identities like \( \sin(A+B) = \sin A \cos B + \cos A \sin B \), and similarly for \( \cos \) transformations.

Step 4: Simplifying the Matrix

Recognize that the structure of the determinant suggests rows are cyclic permutations. This means each row is a version of another row just rotated by constant trigonometric conditions (\(\frac{2\pi}{3}, -\frac{2\pi}{3}\)). This implies that the determinant has a symmetry.

Step 5: Evaluating the Determinant with Symmetry

Given the symmetry and transformation of angles, the determinant of the matrix cancels itself out due to the identical yet offset nature of the rows creating linear dependencies among them.

Step 6: Conclusion

Due to symmetry and linear dependency, the determinant of such a matrix structure is zero. Hence, the determinant is independent of \(\theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics, allowing us to simplify and transform expressions involving angles and trigonometric functions. In this exercise, we utilized such identities to analyze a 3x3 matrix filled with trigonometric terms. For example:

• The sine angle sum identity is given by \[\sin(A + B) = \sin A \cos B + \cos A \sin B. \]
• Similarly, the cosine angle sum identity is \[\cos(A + B) = \cos A \cos B - \sin A \sin B.\]

This helps us express complex trigonometric terms into simpler components. By adopting these identities, we can address the shifted sine and cosine terms in the matrix more effectively. These transformations not only simplify calculations but also reveal underlying symmetries that are crucial for further evaluation. Understanding and applying these identities is key to solving many problems involving trigonometric expressions.

Matrix Symmetry

Matrix symmetry refers to properties that remain unchanged when certain operations are performed on matrices. In our exercise, the matrix exhibits symmetry across its rows. Identifying symmetry can significantly simplify the evaluation of a determinant as it allows us to recognize patterns and predict behavior without calculating manually.In the given 3x3 matrix, each row is a cyclic permutation of the others due to systematic shifts in the trigonometric terms, such as \(\frac{2\pi}{3}\) and \(-\frac{2\pi}{3}\). This is a key feature of symmetry within the matrix, as it shows a structured similarity. Such symmetry indicates that any linear transformation or permutation results in consistent outcomes across different matrix rows.Exploring this concept helps us conclude that the determinant of a symmetric matrix, especially with cyclic permutations like this one, leads to a net zero, as cancellation effects set in. Recognizing symmetry allows us to infer much about the matrix behavior with minimal computation effort, proving the determinant's independence of \( \theta \).

Linear Dependency

Linear dependence occurs when a set of vectors, such as rows or columns of a matrix, can be expressed as a linear combination of each other. In this exercise, linear dependency is a pivotal concept that explains why the determinant evaluates to zero.When solving the determinant of the trigonometric matrix, we identify that each row can be seen as a cyclic permutation of another, revealing linear dependency among them. This means there is no unique solution for the rows, as one row is just a different form of a combination of the others. Thus, the matrix lacks full rank, and its determinant is zero.Understanding linear dependency helps to approach intricate matrix problems efficiently. If the rows of a matrix are dependent, the solutions form planes in space that do not span the entire space. Grasping this concept is essential when working with matrices, as it illuminates solution sets that are otherwise not obvious. The determinant turns out to be constant in this case, further supporting its independence from the variable \( \theta \). Identifying such relationships early can save significant computational effort.