Question

In a triangle ABC, the value of the determinant $\left|\begin{array}{ccc}sin\frac{A}{2}& sin\frac{B}{2}& sin\frac{c}{2}\\ sin(A+B+C)& sin\frac{B}{2}& cos\frac{A}{2}\\ cos\left(\frac{A+B+C}{2}\right)& tan(A+B+C)& sin\frac{C}{2}\end{array}\right|$ is less than or equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. None of these

Answer 1

The correct option is C $\frac{1}{8}$

Let $\Delta =\left|\begin{array}{ccc}sin\frac{A}{2}& sin\frac{B}{2}& sin\frac{c}{2}\\ sin(A+B+C)& sin\frac{B}{2}& cos\frac{A}{2}\\ cos\left(\frac{A+B+C}{2}\right)& tan(A+B+C)& sin\frac{C}{2}\end{array}\right|=\left|\begin{array}{ccc}sin\frac{A}{2}& sin\frac{B}{2}& sin\frac{c}{2}\\ 0& sin\frac{B}{2}& cos\frac{A}{2}\\ 0& 0& sin\frac{C}{2}\end{array}\right|=sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
but $sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\le \frac{1}{8}\therefore \Delta \le \frac{1}{8}$