Can Theta Be Determined from Antenna Field Pattern Equations?

[yungman]

This is part of the antenna field pattern calculation, I want to find out is there any way to find the $\;\theta\;$ of this:

$$\frac 1 N \left[\frac {\sin(N\theta)}{sin\theta}\right]\;\hbox { = 0.7071 where N is an integer bigger than 1}$$

 

[appelberry]

you can use the identity
$\sin N\theta = \sum_{k=0}^N \binom{N}{k} \cos^k \theta\,\sin^{N-k} \theta\,\sin\left(\frac{1}{2}(N-k)\pi\right)$
to obtain a polynomial in $\sin \theta$ which can then be solved for $\theta$. However, you will probably need to solve it numerically for $N>3$.