Let $n \in \mathbb N$ be given and $\phi\in (0,\pi)$ be some angle. Consider the function $$ \frac{\sin(n\phi/2)}{\sin(\phi/2)}. $$ For very small $\phi$ , this function "approaches" the maximum $n^2$. But I do not understand the following argument, taken from The Feynman Lectures, Volume I, chapter 30:
Now we go to the next maximum, and we want to see that it is really much smaller than the first one, as we had hoped. We shall not go precisely to the maximum position, because both the numerator and the denominator of [the above formula] are variant, but $\sin\phi/2$ varies quite slowly compared with $\sin n\phi/2$ when $n$ is large, so when $\sin(n\phi/2) = 1$ we are very close to the maximum. The next maximum of $\sin^2(n\phi/2)$ comes at $n\phi/2 = 3\pi/2$, or $\phi = 3\pi/n$.
I do not understand why the next maximum comes for $\sin(n\phi/2) = 1$? As I see it, as $\sin(\phi/2)$ could take any value, the quotient could realise any value too, so some local maximum could be even before that point?
You may be mixing up local and global maxima. Feynman's argument is that $\sin (\phi/2)$ varies slowly enough that it can be taken as essentially constant on intervals around the peaks of $\sin (n\phi/2)$, so the local extrema of $\sin (n\phi/2)/\sin (\phi/2)$ occur more or less at the same values of $\phi$ as the local extrema of $\sin (n\phi/2)$. Whether these local extrema are also global extrema, of course, depends on the value of $\sin (\phi/2)$ (as you correctly note).