1/sinA+1/sinB+1/sinc>2✔️3 throughout Jensen inequality 

The inequality you provided suggests that the sum of the reciprocals of the sines of angles \( A \), \( B \), and \( C \) is greater than \( 2\sqrt{3} \), written as:

\[
\frac{1}{\sin A} + \frac{1}{\sin B} + \frac{1}{\sin C} > 2\sqrt{3}
\]

To prove this using **Jensen's inequality**, we must recall that Jensen’s inequality applies to convex or concave functions. For this specific case, we need to recognize the convexity or concavity of the reciprocal of the sine function, \( f(x) = \frac{1}{\sin x} \).

### Steps:

1. **Convexity of \( \frac{1}{\sin x} \):**
   The function \( \frac{1}{\sin x} \) is convex in the interval \( (0, \pi) \). This is because its second derivative is positive in this interval, making it a convex function.

2. **Applying Jensen’s Inequality:**
   Given that \( A + B + C = \pi \) (in a triangle), and since \( \frac{1}{\sin x} \) is convex, by Jensen's inequality we can state:
   
   \[
   \frac{1}{\sin A} + \frac{1}{\sin B} + \frac{1}{\sin C} \geq 3 \cdot \frac{1}{\sin\left(\frac{A + B + C}{3}\right)} = 3 \cdot \frac{1}{\sin\left(\frac{\pi}{3}\right)} = 3 \cdot \frac{1}{\frac{\sqrt{3}}{2}} = 2\sqrt{3}
   \]

   Thus, Jensen's inequality confirms that:

   \[
   \frac{1}{\sin A} + \frac{1}{\sin B} + \frac{1}{\sin C} \geq 2\sqrt{3}
   \]

3. **Strict Inequality:**
   Since equality holds when \( A = B = C = \frac{\pi}{3} \), for general triangles where the angles are not all equal, we have the strict inequality:

   \[
   \frac{1}{\sin A} + \frac{1}{\sin B} + \frac{1}{\sin C} > 2\sqrt{3}
   \]

Thus, the inequality is confirmed via Jensen's inequality.

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Would you like more details on any specific part of the explanation?

Here are 5 questions to deepen understanding:
1. What is the significance of a function being convex when applying Jensen's inequality?
2. Can you prove why \( f(x) = \frac{1}{\sin x} \) is convex on \( (0, \pi) \)?
3. How would Jensen's inequality change if we applied it to concave functions?
4. Why does the equality case occur when \( A = B = C = \frac{\pi}{3} \)?
5. How can we use similar inequalities in triangle geometry for different trigonometric functions?

**Tip**: To easily check whether a function is convex, compute its second derivative and verify if it's always non-negative over the interval in question.

1/sinA + 1/sinB + 1/sinC > 2√3 throughout Jensen inequality

Discover how to prove the inequality 1/sinA + 1/sinB + 1/sinC > 2√3 using Jensen's inequality. This solution explains the convexity of the function 1/sin(x) and applies the inequality to a triangle with angles A, B, and C. Suitable for advanced high school students (Grades 11-12).

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