# Maximum value of $\sin A+\sin B+\sin C$ without calculus or Lagrange Multipliers

Let $$A, B, C$$ be a the angles of a triangle. Find the maximum value of $$\sin A+\sin B+\sin C$$ without calculus or Lagrange Multipliers.
My attempt: If we fix $$A$$, then $$B+C$$ is also fixed. We have $$\sin B+ \sin C=2\sin\frac{B+C}{2}\cos\frac{B-C}{2}$$. The maximum value of $$\cos\frac{B-C}{2}$$ is $$1$$ and it is attained when $$B=C$$. What should I do next?

• Hint: Jensen’s inequality. May 7, 2019 at 5:13
• math.stackexchange.com/questions/990418/… May 7, 2019 at 5:17
• I am not interested in a proof by Jensen's inequality because it is calculus related and I already know that proof. May 7, 2019 at 5:29
• @lab bhattaharjee thanks for the link, but it doesn't explain what to do after concluding that $B=C$ and this is where I am stuck. May 7, 2019 at 5:30
• When $B = C$, $A=\pi - 2B$. Now rewrite the equation in terms of $B$. Find the condition on B and then find the minimum value. May 7, 2019 at 6:03

(1) $$\sin A+\sin B+\sin C$$ is the maximum when the area of the quadrilateral $$PQRS$$ is the maximum. (2) With $$A$$ fixed, the area of $$PQRS$$ is the maximum when the area of $$\triangle QRS$$ is the maximum, i.e. when $$R$$ is the farthest point on the arc $$QS$$ from the chord $$QS$$. So, $$R$$ is the midpoint of arc $$QS$$. So, $$B=C$$. (3) The area remains unchanged if $$A$$ and $$B$$ are interchanged. $$PQRS$$ is then an isosceles trapezium inscribed in the semi-circle. (4) Reflect the trapezium in $$PS$$ to form a hexagon inscribed in the unit circle. (5) The area of $$PQRS$$ is equal to the area of $$QRST$$. If it is maximized, $$QRST$$ should also be an isosceles trapezium. SO, $$\angle RSP=\angle QOP$$. Note that $$\angle RSP=\angle QPS=\angle PQT$$. Therefore, $$\triangle OPQ$$ is equilateral. $$\sin A+\sin B+\sin C$$ is the maximum when $$A=B=C=\dfrac{\pi}{3}$$.

• Wow, this is amazing! I hadn't thought about a geometric proof. Could you also please look at my work and tell me how to finish my idea? After getting that $B=C$ I got that $\sin A+\sin B+\sin C=2\sin B(1+\cos B)$ and I don't know how to maximise this. May 7, 2019 at 8:58
• If $f(A,B,C)=\sin A+\sin B+\sin C$. $f(A,B,C)\le f(A,\frac{B+C}{2},\frac{B+C}{2})$. So $f(A,B,C)$ is not the maximum if $B\ne C$. Similarly, $f(A,B,C)$ is not the maximum if $A\ne B$ and $f(A,B,C)$ is not the maximum if $A\ne C$. If $f(A,B,C)$ has a maximum, it must happen when $A=B=C$. May 7, 2019 at 9:08