MAX 1

[rsoldo]

Determine the maximum value of

$$3\cdot\sin\left(x+\frac{\pi}{9}\right)+5\cdot\sin\left(x+\frac{4\pi}{9}\right)$$

for all $x\in\mathbb{R}$.

[zykov]

Subsitute $y=x+\frac{\pi}{9}$ as the new variable.
The rest is easy - either geometrically or trigonometrically.