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i'm looking for relation between angles thats enable the trigonometric vector $u$ to be unitaire i.e $ u = [{\cos}({\theta _1}),{\cos}({\theta _2}),.....{\cos}({\theta _N})]$

and $||u||_2^2 = 1$. where $N>2$

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  • $\begingroup$ Well...there's the obvious $\theta_2 = \frac{\pi}{2} - \theta_1$, and all other $\theta_i$ are $\pi/2$. Can you give any hint of what kind of thing you might mean? $\endgroup$ Mar 9, 2020 at 1:53

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i found that : $\begin{split}R \triangleq \frac{\sin(N \frac{1}{2}d)}{\sin(\frac{1}{2} d)} \\ \sum_{n=0}^{N-1} \cos(a + nd) = \begin{cases} N \cos a & \text{if } \sin(\frac{1}{2}d) = 0 \\ R \cos ( a + (N - 1) \frac{1}{2} d) & \text{otherwise} \end{cases} \\ \sum_{n=0}^{N-1} \sin(a + nd) = \begin{cases} N \sin a & \text{if } \sin(\frac{1}{2}d) = 0 \\ R \sin ( a + (N - 1) \frac{1}{2} d) & \text{otherwise} \end{cases}\end{split}$ so i thinks that for given $N$ and any angle $a$ we can find $d$ numerically ?

2{\cos ^2}({\theta _1}) = 1 + \cos (2{\theta _1})$

so we must find $d$ that verify :

$R(2 - N) = 1$

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