I have a function: $\sin (10.99*x)$
How to find a min-max extent of x where all the function values are positive?
Is brute-force the only solution?
Is there also a possible solution if multiple sine functions of different periods are plotted at the same co-ordinate plane? The question would be, what is the extent of all the x's?
Let's build up to an answer. You will need some basic knowledge of the $\sin$ function.
You probably learned $\sin$ first in degrees: $\sin(0 ^\circ) = 0$, it grows to a maximum of $1$ at $90 ^\circ$, falls back to $0$ at $180 ^\circ$, goes negative and reaches $-1$ at $270 ^\circ$, comes back up to $0$ at $360 ^\circ$ and then repeats.
Serious maths usually uses radians rather than degrees. So, that becomes: $\sin(0) = 0$, it grows to a maximum of $1$ at $\pi /2$, falls back to $0$ at $\pi$, goes negative and reaches $-1$ at $3 \pi / 4$, comes back up to $0$ at $2 \pi$ and then repeats.
So, in radians, it is positive in the range $(0, \pi)$ not inclusive. It is also positive again in the range $(2 \pi, 3 \pi)$ and this repeats every $2 \pi$.
Now, let's look at $sin(2 x)$. Can we deduce its behaviour from that of $sin(x)$. Let's write $2x$ as $y$. When is $\sin(y)$ positive? Well, when $y$ is in the ranges $(0, \pi)$, $(2 \pi, 3 \pi)$, etc. So, if $y$ is in the range $(0, \pi)$ what about $x$? Well $x$ is just half of $y$ so $x$ must be in the ranges $(0, \pi / 2)$, $(\pi, 3 \pi / 2)$, etc.
What about your example? Just do my last example again with $10.99$ in place of $2$. The same except for messier arithmetic.