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

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 a/an (A) equilaterral (B) isosceles (C) right-angled triangle (D) any triangle

Step-by-Step Solution

Answer

The triangle is equilateral (A).

Step 1: Understanding the Problem

We have been given a determinant equation involving angles \(A\), \(B\), and \(C\) of a triangle. We need to determine the type of triangle based on the condition that the determinant is equal to zero.

Step 2: Analyzing the Determinant Condition

The determinant is written as \[\begin{vmatrix}1 & 1 & 1 \ 1+\sin A & 1+\sin B & 1+\sin C \ \sin A+\sin^2 A & \sin B+\sin^2 B & \sin C+\sin^2 C\end{vmatrix} = 0.\]This condition suggests that the rows (or columns) of the matrix are linearly dependent.

Step 3: Simplifying the Determinant

For the determinant of a 3x3 matrix to be zero, one common condition is that two of its columns or rows are identical or multiples of each other. Here, the linearly dependence might indicate a symmetrical property in the values of sine involved, often implying an equilateral characteristic in a triangle context.

Step 4: Exploring the Sine Values for Triangles

If the triangle is equilateral, all the angles are equal, so \(A = B = C\). The sine values would be \(\sin A = \sin B = \sin C\), making the columns of the determinant more likely to satisfy the condition for determinant zero.

Step 5: Reasoning the Triangle Type

We conclude that the condition \(\sin A = \sin B = \sin C\) and symmetry in the matrix elements best fits an equilateral triangle.

Key Concepts

Matrix Determinant

To understand why the determinant of a matrix is important here, let’s start by simplifying what a determinant represents. A determinant is a special number that can be calculated from a square matrix. It gives us important information about the matrix and its properties, like whether it has an inverse or if the rows or columns of the matrix are linearly independent.
When the determinant of a matrix is zero, it indicates a specific characteristic: the rows or columns depend linearly on each other. In the context of angles in a triangle, this linear dependency suggests that some symmetrical property, such as equal angles or sides, might be present, often hinting at a specific type of triangle like an equilateral triangle.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In this problem, trigonometric functions like sine are involved, highlighting the angles of a triangle.
For an equilateral triangle, where all angles are the same ( 180° divided equally to 60°), the sine function values are identical: ext{sin} 60° = ext{sin} 60° = ext{sin} 60° . This identity in sine values is crucial because it points toward the determinant condition being satisfied (zero), implying the angles have this symmetry.
By analyzing the trigonometric functions, and matching them to the properties of equilateral triangles, we understand why the determinant resolves to zero in this scenario.

Properties of Triangles

Triangles have unique properties related to their angles and sides that classify them into categories such as equilateral, isosceles, or right-angled. An equilateral triangle is a special type where all sides and angles are equal. This makes its sine values, used in the determinant, symmetrical or identical.
In an isosceles triangle, two angles are equal, leading to some, but not all, simplification in the determinant, which generally does not suffice the condition of being zero unless in special cases.
Understanding these properties helps identify which type of triangle fits the determinant’s zero condition. Given the matrix in the exercise, the conclusion is that only an equilateral triangle satisfies the symmetry and linear dependency requirement indicated by a determinant of zero.