Mathematics

The key is to recognize that a sum of sines can often be factored using sum-to-product formulas. Group the first and third terms strategically: $\sin x + \sin 9x$ combines into a product form, which you can then factor together with $\sin 5x$. This transforms the equation into a product of simpler trigonometric expressions, each of which you can solve independently. Counting solutions carefully across the interval $(0, 2\pi)$ while avoiding double-counting shared roots will give you the total.

The key is to recognize that a sum of sines can often be factored using sum-to-product formulas. Group the first and third terms strategically: $\sin x + \sin 9x$ combines into a product form, which you can then factor together with $\sin 5x$. This transforms the equation into a product of simpler trigonometric expressions, each of which you can solve independently. Counting solutions carefully across the interval $(0, 2\pi)$ while avoiding double-counting shared roots will give you the total.