Question

In any $△ABC$, $\Sigma \left(\frac{{\sin }^{2}A+\sin A+1}{\sin A}\right)$ is always greater than :

• A. $9$
• B. $3$
• C. $27$
• D. none of these

Answer 1

Answer: (A)

$9$

Let $y=\Sigma \left(\frac{{\sin }^{2}A+\sin A+1}{\sin A}\right)=\Sigma [1+(\sin A+\frac{1}{\sin A})]$

Now we know in any triangle all $\sin A,\sin B,\sin C$ are positive.

Thus using $A.M\ge G.M\Rightarrow \sin A+\frac{1}{\sin A}\ge 2$

Hence $y\ge \Sigma (1+2)=9$