Question

Assertion :The minimum value of the expression $\sin \alpha +\sin \beta +\sin \gamma$ where $\alpha ,\beta ,\gamma$ are real numbers such that $\alpha +\beta +\gamma =\pi$ is negative. Reason: If $\alpha +\beta +\gamma =\pi$,then $\alpha ,\beta ,\gamma$ are the angles of a triangle.

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

The minimum value of the sum can be -3 provided $\sin \alpha =\sin \beta =\sin \gamma =-1$

$\Rightarrow \alpha =(4l-1)\pi /2,$

$\beta =(4m-1)\pi /2,$

$\gamma =(4n-1)\pi /2$

Now $\alpha +\beta +\gamma =\pi \Rightarrow \left[4\right(l+m+n)-3]\pi /2=\pi$
$\Rightarrow 4(l+m+n)=5$

which is not possible as l, m, n are integers.
1. minimum value can not be -3.

But for $\alpha =3\pi /2,\beta =3\pi /2,\gamma =2\pi ,\alpha +\beta +\gamma =\pi$ and $\sin \alpha +\sin \beta +\sin \gamma =-2$

So $\sin \alpha +\sin \beta +\sin \gamma$ can have negative values and thus the minimum value of the sum is negative proving that statement-1 is correct.

But the statement-2 is false as $\alpha +\beta +\gamma =\pi$ for $\alpha =\beta =2\pi /3,\gamma =-2\pi$ which are not the angles of a triangle.