Question

Assertion :The minimum value of the expression $\sin \alpha +\sin \beta +\sin \gamma$ where $\alpha ,\beta ,\gamma$ are real number such that $\alpha +\beta +\gamma =\pi$, is negative because- Reason: $\alpha ,\beta ,\gamma$ are 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

We know that $\sin \theta$ is minimum at $\theta =-\frac{\pi }{2}$
Now,

let, $\alpha =-\frac{\pi }{2}\text{and}\beta =-\frac{\pi }{2}$
then,
$\alpha +\beta +\gamma =\pi \left[\text{given}\right]\Rightarrow -\frac{\pi }{2}-\frac{\pi }{2}+\gamma =\pi \therefore \gamma =2\pi$

$\text{minimum value of}\sin \alpha +\sin \beta +\sin \gamma =\sin (-\frac{\pi }{2})+\sin (-\frac{\pi }{2})+\sin \left(2\pi \right)=-1-1+0=-2$

$\therefore \text{minimum value of}\sin \alpha +\sin \beta +\sin \gamma \text{is negative}$

Now,

consider $\alpha ,\beta ,\gamma$ are not angles of a triangle
then we can take,
$\alpha =\beta =\gamma =-\frac{\pi }{2}\Rightarrow \alpha +\beta +\gamma =\frac{-3\pi }{2}\therefore \alpha =\beta =\gamma =-\frac{\pi }{2}\text{are not angles of a triangle in this case}$
Then,

$\text{minimum value of}\sin \alpha +\sin \beta +\sin \gamma$
$=\sin (-\frac{\pi }{2})+\sin \left(\frac{-\pi }{2}\right)+\sin \left(\frac{-\pi }{2}\right)$
$=-1-1-1=-3$

In this case, $\alpha ,\beta ,\gamma$ are not angles of a triangle
but minimum value of $\sin \alpha +\sin \beta +\sin \gamma$ is negative

Hence,
Assertion is correct but Reason is incorrect