Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A, B, C\) are the angles of a triangle and \(\left|\begin{array}{ccc}1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+\sin ^{2} A & \sin B+\sin ^{2} B & \sin C+\sin ^{2} C\end{array}\right|=0\) then the triangle is \(\mathrm{a} /\) an (A) equilaterral (B) isosceles (C) right-angled triangle (D) any triangle

Step-by-Step Solution

Answer

The triangle is equilateral.

Step 1: Understand the Problem

We need to determine the type of triangle based on the given determinant condition. The determinant involves the angles (and their trigonometric functions) of the triangle.

Step 2: Identify the Matrix Structure

We have a 3x3 matrix formed by the constants 1, terms involving \(1 + \sin A\), \(1 + \sin B\), \(1 + \sin C\) and terms involving \(\sin A + \sin^2 A\), \(\sin B + \sin^2 B\), \(\sin C + \sin^2 C\). The determinant of this matrix is given to be zero.

Step 3: Apply Properties of Determinants

The determinant being zero suggests that the columns (or rows) are linearly dependent, which implies a special relationship between the angles \(A\), \(B\), and \(C\).

Step 4: Check for Symmetry

Since the determinant involves symmetric functions of \(A\), \(B\), and \(C\), this often leads to conditions of equality or specific angle values that define certain types of triangles.

Step 5: Determine Special Triangle Conditions

Given that the determinant is zero with symmetrical structure, the simplest scenario is when all angles are equal, suggesting triangle could be equilateral. Alternatively, symmetrical determinants might hold when two angles are equal, indicating an isosceles triangle.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are foundational in understanding triangles. They relate angles of triangles to the lengths of their sides. In our exercise, sine is used extensively.

The sine function, denoted as \( \sin \theta \), gives the ratio of the opposite side to the hypotenuse in a right-angled triangle. However, it also applies more generally in any triangle via the sine rule.

• Sine is defined for all real numbers, with a range from -1 to 1.
• When angles are part of a triangle, their sine values help determine the nature of the triangle.
• For example, in an equilateral triangle, the sine of angles is same.

Trigonometric functions thus provide insights into the relationship between the triangle’s angles and its sides, crucial in solving the determinant problem in the exercise.

Matrix Determinants

A matrix determinant is a scalar value that can be calculated from a square matrix. Determinants have many important properties and applications, particularly in solving systems of linear equations.

In this scenario, we have a 3x3 matrix, and its determinant being zero indicates linear dependence among its rows or columns. This particular determinant is based on the structures of sinusoids related to a triangle's angles.

• Determinants help identify whether a matrix is invertible; a zero determinant means it isn't.
• For a 3x3 matrix like in the exercise, the computation involves finding products of elements in a specific pattern, which here simplifies to checking angle symmetry.
• Application of properties of determinants may reveal geometric configurations of the triangles through the involved angles.

Understanding matrix determinants is pivotal, as it flags key relationships in the given triangle, leading us towards concepts like symmetry in angles, which determine the triangle's type.

Linear Dependence in Matrices

Linear dependence in matrices is a concept that arises when rows or columns of a matrix can be expressed as linear combinations of each other.

This property significantly impacts the structure and type of solutions we can obtain from a matrix equation. In our problem, the linear dependence implied by the determinant being zero suggests that the sine expressions involving angles \(A, B,\) and \(C\) are not independent.

• Linear dependence is a reason for a zero determinant, indicating some rows or columns of the matrix are directly related.
• In our context, symmetry suggests possible equality among angles.
• This linear dependence prompts us to observe equalities, leading to possibilities like equilateral triangles when all angles are equal.

Recognizing linear dependence enables us to deduce important properties about the triangle, refining our understanding of its potential configurations such as being equilateral or isosceles.