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

Question

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

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Understand the Matrix

The problem asks us to find the determinant of a 3x3 matrix, where each element is a trigonometric function. Specifically, the matrix depends on the angle \( \theta \). By recognizing the symmetry and periodicity in trigonometric functions, we can simplify the matrix components.

Step 2: Simplify the Trigonometric Expressions

Use known trigonometric identities to simplify each component. For instance, \[\sin \left(\theta + \frac{2\pi}{3}\right) = \sin \theta \cos \frac{2\pi}{3} + \cos \theta \sin \frac{2\pi}{3} = -\frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta\] and similarly for other sine and cosine terms using sum and difference formulas.

Step 3: Calculate the Determinant

A determinant of a 3x3 matrix is calculated using the formula: \[\mathrm{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]where \(a, b, c\) are elements from the first row, and \(d, e, f, g, h, i\) are the corresponding elements. Substitute the simplified trigonometric expressions and compute the determinant explicitly.

Step 4: Evaluate Determinant for Identity

After substituting and simplifying, observe if the determinant remains constant or changes with \( \theta \). This involves checking if the terms cancel out or follow a specific trigonometric identity, leading to a constant determinant value or zero.

Step 5: Conclude the Result

Through computation and comparison, it's evident that the determinant simplifies to zero for all \( \theta \). This is due to the trigonometric identities and symmetry in the algorithm, which cancel out terms and lead to a zero determinant.

Key Concepts

trigonometric identities

Trigonometric identities are essential tools in simplifying expressions involving trigonometric functions, especially when angles are involved. These identities allow us to convert complex trigonometric expressions into simpler forms, making calculations more manageable.

• **Basic Identities:** These include the Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), which are foundational in manipulating expressions.
• **Sum and Difference Formulas:** These are used to express trigonometric functions of sums or differences of angles in terms of the functions of the individual angles, such as \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \).
• **Double Angle Identities:** Functions like \( \sin 2\theta \) can be rewritten using such identities, for instance, \( \sin 2\theta = 2 \sin \theta \cos \theta \).

When we apply these identities in our problem, they help us rewrite complex functions as combinations of simpler sine and cosine terms. This simplification is crucial for determining if expressions cancel out or simplify further.

determinants in matrices

Determinants are a crucial concept in linear algebra and play an important role in understanding the properties of matrices. For a 3x3 matrix, the determinant provides a scalar value that can indicate several properties of the matrix, including invertibility.
The formula to compute the determinant of a 3x3 matrix \(\left| \begin{array}{ccc}a & b & c \d & e & f \g & h & i \\end{array} \right| \) is: \[ det(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \]

• This formula involves taking products among the elements of the matrix in a specific cross-multiplication pattern.
• The determinant can indicate if a matrix has inverses; if the determinant is zero, the matrix is singular, which means it does not have an inverse.

In our exercise, the determinant's calculation helps us determine if the matrix behavior is consistent across trigonometric variations, ultimately simplifying to zero.

matrix simplification

Simplifying a matrix, especially one involving trigonometric functions, involves breaking down each component using trigonometric identities. This process can be crucial in making calculations less cumbersome and more understandable.
To simplify a matrix, consider the following steps:

• **Identify Simplifiable Parts:** Look for terms that can be simplified using known mathematical identities or properties.
• **Apply Identities:** Use trigonometric identities to express complex trigonometric expressions in more straightforward forms.
• **Substitute Back:** Once simplified, substitute these simplified terms back into the matrix to make calculations easier.

By applying these principles to our matrix with trigonometric elements, it becomes feasible to perform determinant calculations and evaluate special properties like symmetry.

trigonometric functions in linear algebra

The presence of trigonometric functions within matrices introduces an additional layer of complexity to linear algebra problems. Understanding how these functions interact within the matrix is important for correctly solving such problems.
In linear algebra, trigonometric functions can couple with matrix operations in unique ways:

• **Periodic Nature:** Trigonometric functions have periodic properties, meaning they repeat values over a specific interval, which can influence matrices by introducing repeating patterns.
• **Symmetry:** Symmetry is often observed due to the periodic and even or odd properties of sine and cosine functions, and can significantly simplify the determinant and other matrix operations.
• **Analytical Solutions:** Using trigonometric identities and functions allows for analytical solutions, where patterns and repeats help predict behavior without computationally intensive calculations.

The study of this interplay assists in conceptualizing consistent solutions and identifying determinants or other matrix invariants that do not change with specific transformations or angle variations.