Question

The maximum value of $\frac{a\cos A+b\cos B+c\cos C}{a+b+c}$ , where a, b, c and A, B, C have their usual meaning, is

• A. $\sin A$
• B. $4\sin A/2\sin B/2\sin C/2$
• C. $\sin C/2$
• D. None of these

Answer 1

Answer: (B)

$4\sin A/2\sin B/2\sin C/2$

$\frac{a\cos A+b\cos B+c\cos C}{a+b+c}willbe\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R\left(sineformula\right)=\frac{R(\sin 2A+\sin 2B+\sin 2C)}{2R(\sin A+\sin B+\sin C)}=\frac{4\sin A\sin B\sin C}{2\left(4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\right)}=4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$