TITLE: Find the maximum value of $\sin (A) + \sin( 2A )+ \sin (3A)$
QUESTION [5 upvotes]: From the graph, it seems that the answer is 2.5. Can someone please help me to solve this problem manually by using trigonometric equations.

REPLY [2 votes]: According to your graph, the maximum seems to appear "close" to $A=\frac \pi 5$. Develop the expression as a Taylor series to get for the function
$$\sqrt{\frac{25}{8}+\frac{11 \sqrt{5}}{8}}+\frac{1}{2} \left(A-\frac{\pi
   }{5}\right)-\frac{1}{4} \sqrt{5 \left(85+22 \sqrt{5}\right)} \left(A-\frac{\pi
   }{5}\right)^2+O\left(\left(A-\frac{\pi }{5}\right)^3\right)$$ Ignoring the higher order terms, the derivative cancels for
$$A=\frac \pi 5+\frac{1}{\sqrt{5 \left(85+22 \sqrt{5}\right)}}\approx 0.666924$$ and for this value the truncated series gives a maximum of
$$\frac{\sqrt{5}+5 \sqrt{6670+2970 \sqrt{5}}}{20 \sqrt{85+22 \sqrt{5}}}\approx 2.49955$$
The exact solution of the problem is $(0.667291,2.49961)$. Not too bad for a solution obtained without solving any equation beside a linear one.