# trigonometric relationship to get unitaire vector

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 $$N>2$$

• 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? Mar 9, 2020 at 1:53

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 ?
so we must find $$d$$ that verify :
$$R(2 - N) = 1$$