Given that in the triangle
ABC,sin3A+sin3B+sin3C=3sinAsinBsinC, we need to determine the value of the determinant:
||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=, 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:
||To compute this determinant, notice that each row is identical. A determinant with identical rows is zero. Therefore:
||=0