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actual evaluation – Number of zeros of a sine polynomial

Let $(a_i)_{i=1}^{n}$ breathe non-negative, then what number of actual zeros does the sine polynomial

$$

p(theta) := a_1sin(theta) + a_2sin(2theta) + cdots + a_nsin(ntheta), quad theta in [0,2pi)

$$

have?

Furthermore:

- Clearly $theta = 0,pi$ are zeros of $p$. What standards for $(a_n)$ live for these to breathe the
**solely**zeros.

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