Question

In any $\Delta ABC,\Pi (\frac{{\sin }^{2}A+\sin A+1}{\sin A})$ is always greater than

• A. $9$
• B. $3$
• C. $27$
• D. $81$

Answer 1

Answer: (A), (B), (C)

$9$
$27$
$3$

$\left(\frac{si{n}^{2}A+sinA+1}{sinA}\right)\left(\frac{si{n}^{2}B+sinB+1}{sinB}\right)\left(\frac{si{n}^{2}C+sinC+1}{sinC}\right)$
$(sinA+\frac{1}{sinA}+1)(sinB+\frac{1}{sinB}+1)(sinC+1+\frac{1}{sinC})$
$\frac{1}{x}+x\ge 2>0$
$1+\frac{1}{x}+x\ge 3$
$\therefore$ the whole term is greater than 27