Question

Assertion :In any triangle $ABC;a\cos A+b\cos B+c\cos C\le s$ Reason: In any triangle $ABC;\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Since $L.H.S$ of the inequality is reason is symmetric function of sines of the angles of the triangle, its maximum values is attained.

When $A=B=C={60}^0$ and hence the reason is true.

Assertion is true if

$2R(\sin A\cos A+\sin B\cos B+\sin C\cos C)\le R(\sin A+\sin B+\sin C)$

or if $4\sin A\sin B\sin C\le 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$ (From conditional identities)

or if $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$

which is true by reason, hence assertion is true.