$\mbox{Euler Formula}$ \begin{aligned} \sin\theta &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \\ \cos\theta &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... \\ e^z &= 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + ... = \sum_{k=0}^{\infty} \frac{z^k}{k!} \\ e^{i\theta} &= 1 + i\theta + \frac{{(i\theta)}^2}{2!} + \frac{{(i\theta)}^3}{3!} + \frac{{(i\theta)}^4}{4!} + \frac{{(i\theta)}^5}{5!} ... \\ e^{i\theta} &= 1 + i\theta - \frac{{\theta}^2}{2!} - \frac{{i\theta}^3}{3!} + \frac{{\theta}^4}{4!} + \frac{{i\theta}^5}{5!} - \frac{\theta^6}{6!}...\\ e^{i\theta} &= (1 - \frac{{\theta}^2}{2!} + \frac{{\theta}^4}{4!} - \frac{\theta^6}{6!} + ...) + i(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - ...)\\ e^{i\theta} &= \sum_{k=0}^{\infty} \frac{\theta^{2k}}{2k!} + \sum_{k=0}^{\infty} \frac{{i\theta}^{2k+1}}{(2k+1)!} \\ e^{i\theta} &= \cos\theta + i\sin\theta \\ \mbox{Other formula from Euler forumla} \\ e^{i\theta} &= \cos\theta + i\sin\theta \qquad &(1)\\ e^{-i\theta} & =\cos\theta - i\sin\theta \quad (\theta = -\theta) \qquad &(2)\\ \Rightarrow 2\cos \theta & = e^{i\theta} + e^{-i\theta} \qquad (1) + (2) \\ \Rightarrow \color{red}{\cos \theta} &= \frac{e^{i\theta} + e^{-i\theta}}{2} \\ \Rightarrow 2i\sin \theta & = e^{i\theta} - e^{-i\theta} \qquad (1) - (2) \\ \Rightarrow \color{red}{\sin \theta} & = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\ \end{aligned}