analysis - All Partial sums of two given sequences are bounded by a positive constant

Let $\theta \in \mathbb R$ be a non-integer multiple of $2\pi$. Prove that the sequences $(\sin(n\theta))_{n \in \mathbb N}$ and $(\cos(n\theta))_{n \in \mathbb N}$ verify $|S_N|\leq K$ where $K>0$ and $S_N=a_1+...+a_N$ for a given sequence $(a_n)_{n \in \mathbb N}$.

I began with the sequence involving the cosine, I suppose that the other case is analogue. I tried to express $\cos(n\theta)$ in its exponential form. Then $\sum_{n=1}^N \cos(n\theta)= \sum_{n=1}^N \frac {1} {2}(e^{in\theta}+e^{-in\theta})$. The second member of the equation can be separated into $\frac {1} {2} (\sum_{n=1}^N e^{in\theta}+ \sum_{n=1}^N e^{-in\theta})$. Both of these are the first $N$ terms of two geometric series. So $\frac {1} {2} (\sum_{n=1}^N e^{in\theta}+ \sum_{n=1}^N e^{-in\theta})=\frac {1} {2} (\frac {e^{i(N+1)\theta}-e^{i\theta}} {e^{i\theta}-1} + \frac {e^{-i(N+1)\theta}-e^{-i\theta}} {e^{-i\theta}-1})$. Well, I know that the denominator is never $0$ by the hypothesis we have on $\theta$. I've been fighting with this last term but I don't get to something nice. Am I doing something wrong?

Answer

Note that $$S_N=\Re\left(\sum_{n=1}^N\mathrm e^{\mathrm in\theta}\right)\quad\text{or}\quad S_N=\Im\left(\sum_{n=1}^N\mathrm e^{\mathrm in\theta}\right),$$ and that $$\sum_{n=1}^N\mathrm e^{\mathrm in\theta}=\frac{\mathrm e^{\mathrm i\theta}-\mathrm e^{\mathrm i(N+1)\theta}}{1-\mathrm e^{\mathrm i\theta}}.$$ Furthermore, for every complex number $z$, $|\Re(z)|\leqslant|z|$ and $|\Im(z)|\leqslant|z|$, hence $$|S_N|\leqslant\left|\frac{\mathrm e^{\mathrm i\theta}-\mathrm e^{\mathrm i(N+1)\theta}}{1-\mathrm e^{\mathrm i\theta}}\right|\leqslant K_\theta,$$ with $$K_\theta=\frac2{|1-\mathrm e^{\mathrm i\theta}|}.$$