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Maths-

General

Easy

Question

In a triangle $ABC$

$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8}$
$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} < \frac{1}{8}$
$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} > \frac{1}{8}$
$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \geq \frac{1}{8}$

hint Hint:

As we know that GM $less or equal than$ AM, from this we can write that $open parentheses sin squared A over 2. sin squared B over 2. sin squared C over 2 close parentheses to the power of 1 third end exponent less or equal than fraction numerator sin squared A over 2 plus sin squared B over 2 plus sin squared C over 2 over denominator 3 end fraction$ . Also, $sin squared A over 2 plus sin squared B over 2 plus sin squared C over 2 greater or equal than 3 over 4$ from both these we can determine the true statement from the following.

The correct answer is: $sin invisible function application A over 2 sin invisible function application B over 2 sin invisible function application C over 2 less or equal than 1 over 8$

We know that GM $less or equal than$ AM,
$open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than fraction numerator sin squared begin display style A over 2 end style plus sin squared begin display style B over 2 end style plus sin squared begin display style C over 2 end style over denominator 3 end fraction A l s o comma space sin squared A over 2 plus sin squared B over 2 plus sin squared C over 2 greater or equal than 3 over 4 open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than 1 third cross times 3 over 4 rightwards double arrow sin squared A over 2. sin squared B over 2. sin squared C over 2 less or equal than open parentheses 1 fourth close parentheses cubed rightwards double arrow open parentheses sin A over 2. sin B over 2. sin C over 2 close parentheses squared less or equal than 1 over 64 rightwards double arrow sin A over 2. sin B over 2. sin C over 2 less or equal than 1 over 8$

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