Problem 88 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The determinant \(\left|\begin{array}{ccc}\sin x & \sin y & \sin z \\ \cos x & \cos y & \cos z \\ \cos ^{3} x & \cos ^{3} y & \cos ^{3} z\end{array}\right| ; 0

Step-by-Step Solution

Answer

The answer is (D) x+y+z=\pi/2.

Step 1: Recognize the Determinant Structure

We have a 3x3 matrix with specific trigonometric functions in each row. Recognize that the solution involves finding conditions under which the determinant of this matrix equals zero.

Step 2: Conditions for Zero Determinant

A determinant of a matrix is zero if its rows (or columns) are linearly dependent. For trigonometric functions, this can happen if the angles involve complementary (or supplementary) relations caused by trigonometric identities.

Step 3: Analyze Potential Trigonometric Identities

Given the angles in the matrix and their trigonometric terms, analyze trigonometric complementary identity conditions where an expression equals zero. For trigonometric functions, a common condition placing these functions under dependency is when their sum equals a special angle, such as pi/2 or pi.

Step 4: Check Options Against Identity

Given (A) x=y, (B) y=z, (C) z=x, and (D) x+y+z=pi/2, check these against known trigonometric identities. Notice that option D relates the angle sum to pi/2, representing orthogonal complements critical to zero joint determinant conditions.

Step 5: Verify Option (D)

Confirm that (D), which states x+y+z equals pi/2, utilizes the important identity sin(pi/2 - theta)=cos(theta) to establish a linear relationship between rows, satisfying the condition for a zero determinant.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are crucial in mathematics, especially when dealing with angles and circles. They help describe the relationships between the angles and sides of triangles. In this context, the determinant involves the functions: \(\sin(x)\), \(\cos(x)\), and \(\cos^3(x)\).
These functions are:

• Sine Function: Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine Function: Represents the ratio of the adjacent side to the hypotenuse.
• Leading to Powers: \(\cos^3(x)\) simply indicates that cosine of \(x\) is raised to the third power, affecting its amplitude and characteristics.

Functions are cyclical, with sine starting at zero and reaching its maximum at 90 degrees (\(\pi/2\)), while cosine starts at one. Understanding these functions helps in analyzing patterns and functional dependencies in matrices, especially when solving for determinants with zero values.

Matrix Linear Dependency

A matrix's determinant is zero if its rows or columns are linearly dependent. This means they are multiples of each other or can be expressed as a combination of others. In simpler terms, no new information is added by one row/column beyond what the others already provide. For the given 3x3 matrix:

• First Row: \([\sin(x), \sin(y), \sin(z)]\)
• Second Row: \([\cos(x), \cos(y), \cos(z)]\)
• Third Row: \([\cos^3(x), \cos^3(y), \cos^3(z)]\)

Linear dependency here may arise if these rows become proportional or related through specific trigonometric identities. For example, if angles relate such that their trigonometric expressions equate to constants or zero values, dependency ensues. Recognizing such situations is vital for determining when a matrix will have a determinant of zero, effectively simplifying complex matrix problems.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions. They help simplify and solve equations with trigonometric terms. These identities often rely on angle properties like complements or symmetry. An important identity is \(1 - \sin^2(x) = \cos^2(x)\). This particular identity hints towards complementary angles where certain angles sum to \(\pi/2\) or \(\pi\).
A relevant identity here is the complementary angle identity:

• Complementary Identity: \(\sin(\pi/2 - \theta) = \cos(\theta)\)

This means when angles satisfy: \(x+y+z=\pi/2\), the corresponding sine and cosine terms from the determinant can result in zero through dependencies.
Therefore, checking which conditions fulfill identities like these is essential when examining options that lead to zero determinant matrices. Recognizing and applying identities helps reduce complexity and derive solutions swiftly, as in identifying that the combination of angles ensuring the determinant equals zero when their sum is \(\pi/2\).