In $ \triangle ABC $, with usual notations, $ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \quad \text{and} \quad 2s \text{ is the perimeter of the triangle. Find the value of } s. $ Then the value of $ s $ is:

Question

For the function $f(x) = \sin|x| - |x|$, $x \in \mathbb{R}$, consider the following statements:
Statement I: $f$ is differentiable for all $x \in \mathbb{R}$.
Statement II: $f$ is increasing in $(-\pi, -\pi/2)$.
In the light of the above statements, choose the correct answer from the options given below:

Answer 1

Given the equation: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \], where $ 2s $ is the triangle's perimeter. Applying the sine rule and relationships between sides and angles, along with the semi-perimeter formula, and solving based on the given conditions, yields the result: \[ s = 3b \]. Therefore, the correct answer is $ 3b $.

In $ \triangle ABC $, with usual notations, $ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \quad \text{and} \quad 2s \text{ is the perimeter of the triangle. Find the value of } s. $ Then the value of $ s $ is:

Show Hint

The Correct Option is C

Solution and Explanation

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