Problem 68 - A Complete Resource Book in Mathematics for JEE Main

Question

If the system of equations \(x \sin \alpha+y \sin \beta+z \sin \gamma=0, x \cos \alpha+y \cos \beta+z \cos \gamma\) \(=0, x+y+z=0\), where \(\alpha, \beta, \gamma\) are angles of a triangle, have a non-trivial solution, then the triangle must be (A) isosceles (B) equilateral (C) right angled (D) None of these

Step-by-Step Solution

Answer

The triangle must be equilateral (Option B).

Step 1: Understanding the Problem

We are given a set of linear homogeneous equations. For these equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero. The given angles \(\alpha, \beta, \gamma\) are angles of a triangle, which implies \(\alpha + \beta + \gamma = \pi\) radians (or 180 degrees).

Step 2: Forming the Coefficient Matrix

The coefficient matrix from the system of equations is: \[A = \begin{bmatrix} \sin \alpha & \sin \beta & \sin \gamma \ \cos \alpha & \cos \beta & \cos \gamma \ 1 & 1 & 1 \end{bmatrix} \]

Step 3: Determinant of the Coefficient Matrix

Calculate the determinant of matrix \(A\). A non-trivial solution exists if and only if the determinant is zero: \[\text{Det}(A) = \begin{vmatrix} \sin \alpha & \sin \beta & \sin \gamma \ \cos \alpha & \cos \beta & \cos \gamma \ 1 & 1 & 1 \end{vmatrix} = 0\] Expanding the determinant along the third row, \(\text{Det}(A) = \sin \alpha (\cos \beta - \cos \gamma) - \sin \beta (\cos \alpha - \cos \gamma) + \sin \gamma(\cos \alpha - \cos \beta) \).

Step 4: Analyzing the Determinant Expression

To solve \(\text{Det}(A) = 0\) and equate it with trigonometric identities, note that for angles in triangles, special conditions exist when two angles are equal due to symmetry and angle sum properties. Thus, an equilateral triangle where \(\alpha = \beta = \gamma\) satisfies the condition.

Step 5: Verifying Triangle Type

Substituting \(\alpha = \beta = \gamma\) in the determinant confirms that it's equilateral. Also, note that all angles in an equilateral triangle are \(\frac{\pi}{3}\) and hence they satisfy the condition \(\sin \alpha = \sin \beta = \sin \gamma\) and \(\cos \alpha = \cos \beta = \cos \gamma\). Thus, the determinant becomes zero.

Key Concepts

Determinant of Matrix

In the context of solving a system of linear equations, the determinant of a matrix is a crucial tool. It helps us understand whether a system has a unique solution, no solution, or infinitely many solutions.
A matrix is often composed of coefficients from a system of equations, and its determinant is a special number calculated from its elements. For a set of three equations involving three variables, such as the ones in this exercise, we can form a 3x3 matrix from the coefficients of the variables.
To determine if a system has a non-trivial solution (other than the trivial solution where all variables are zero), the determinant of its coefficient matrix must be zero. This is because a zero determinant indicates that the rows or columns of the matrix are linearly dependent, meaning there are combinations of the variables that can satisfy all equations simultaneously without all variables being zero.
The determinant of a matrix \(A\) can be calculated in various ways depending on its size. For a 3x3 matrix, one common method is the cross multiplication tactic, expanding along any row or column and combining its elements with the appropriate minors and their sign variations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They play a key role in simplifying expressions and solving equations that include trigonometric terms.
In this exercise, we are dealing with angles \(\alpha, \beta, \gamma\) that belong to a triangle. Such angles need to satisfy the identity \(\alpha + \beta + \gamma = \pi\) (or 180 degrees in degrees), as this is the sum of angles in any triangle.
We use trigonometric identities to manipulate these angles and demonstrate equivalences or relationships. For example, knowing the sum of angles, we can derive identities for \(\sin\) and \(\cos\) using known identities like the sine rule, cosine rule, or others. In the equation solution, knowing these identities allows us to adjust the expression of the determinant to convenient forms or even verify if one particular state of the angles, such as being equal, satisfies the condition needed for the solution.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length and, as a result, all three angles are equal. Each angle of an equilateral triangle is \(\frac{\pi}{3}\) radians, or 60 degrees, reflecting its perfect symmetry.
In problems involving mathematical proofs or identities, equilateral triangles often come into play because of their symmetric properties which simplify many expressions.
If we take a closer look at the system of equations given in this problem, there is a significant advantage in setting \(\alpha = \beta = \gamma\). When an equilateral triangle is included in the consideration, the trigonometric values for \(\sin\) and \(\cos\) functions are the same for each angle. This uniformity can often reduce complex expressions into simpler forms.
In the provided exercise, such uniform angles cause both \(\sin\) and \(\cos\) of the angles to align perfectly, resulting in a coefficient matrix determinant of zero. This effectively suggests that, under the circumstances of the problem, the triangle must be equilateral for the system to have a non-trivial solution.