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This question already has an answer here:

Can anyone help me how to prove $\sin A + \sin B + \sin C \leq \frac{3}{2} \cdot \sqrt[2]{3} $

I have idea use Jensen but how to use it here?

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marked as duplicate by user26486, graydad, Hippalectryon, saz, user149792 Mar 3 '15 at 19:57

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ If $A=B=C=90^\circ$ then $\sin A+\sin B+\sin C=3>\dfrac 3 2 \sqrt{3}$. If you intended some hypothesis on $A,B,C$, e.g. they are the three angles in a triangle, then that should be stated. $\endgroup$ – Michael Hardy Mar 3 '15 at 15:30
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This question has already been answered, using Jensen's inequality, here: https://math.stackexchange.com/a/990423/195938

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