UPSC NDA I 2024 Mathematics Paper

Given that in the triangle ABC,sin‌3A+sin‌3B+sin‌3C=3sin‌Asin‌Bsin‌C, we need to determine the value of the determinant:
|

a	b	c
b	c	a
c	a	b

|
From the given trigonometric identity, we know that sin‌A,sin‌B, and sin‌C satisfy a specific relationship due to the nature of the angles in triangle ABC. In this context, the identity sin‌3A+sin‌3B+sin‌3C=3sin‌Asin‌Bsin‌C often arises in cases where A=B=C=‌

π

3

, thus making triangle ABC equilateral.
In an equilateral triangle, the sides a,b, and c are equal. Hence, let's denote the side length of the equilateral triangle as a (i.e., a=b=c ). Substituting these in the determinant, we get:
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a	a	a
a	a	a
a	a	a

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To compute this determinant, notice that each row is identical. A determinant with identical rows is zero. Therefore:
|

a	b	c
b	c	a
c	a	b

|=0