Trigonometry

Remember

\[\begin{align*} \sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta\\ \sin(\alpha - \beta) &= \sin\alpha \cos\beta - \cos\alpha \sin \beta\\ \cos(\alpha + \beta) &= \cos\alpha \cos\beta - \sin\alpha \sin\beta\\ \cos(\alpha - \beta) &= \cos\alpha\cos\beta + \sin\alpha \sin\beta\\ \end{align*}\]

TODO

\[\begin{align*} \cot \theta &= \frac{1}{\tan\theta}\\ \sec \theta &= \frac{1}{\cos\theta}\\ \csc \theta &= \frac{1}{\sin\theta}\\ \frac{e^{i\theta} + e^{-i\theta}}{2} &= \cos\theta\\ \frac{e^{i\theta} - e^{-i\theta}}{2i} &= \sin\theta\\ sinc(x) &= \frac{\sin(\pi x)}{ \pi x}\\ \sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta\\ \sin(\alpha - \beta) &= \sin\alpha \cos\beta - \cos\alpha \sin \beta\\ \cos(\alpha + \beta) &= \cos\alpha \cos\beta - \sin\alpha \sin\beta\\ \cos(\alpha - \beta) &= \cos\alpha\cos\beta + \sin\alpha \sin\beta\\ \sin \alpha \cos \beta &= \frac{1}{2}\big( \sin(\alpha + \beta) + \sin(\alpha -\beta) \big)\\ \cos\alpha \sin\beta &= \frac{1}{2}\big( \sin(\alpha + \beta) - \sin(\alpha - \beta) \big)\\ \cos\alpha \cos\beta &= \frac{1}{2}\big( \cos(\alpha + \beta) + \cos(\alpha - \beta) \big)\\ \sin\alpha \sin\beta &= -\frac{1}{2}\big( \cos(\alpha + \beta) - \cos(\alpha - \beta) \big)\\ A\sin\alpha + B\cos\alpha &= \sin(\alpha+\varphi)\quad\textrm{where }\tan\varphi = \frac{B}{A}\\ A\sin\alpha - B\cos\alpha &= \sin(\alpha - \varphi)\quad\textrm{where }\tan\varphi = \frac{B}{A}\\ A\cos\alpha - B\sin\alpha &= \cos(\alpha + \varphi)\quad\textrm{where }\tan\varphi = \frac{B}{A}\\ A\cos\alpha + B\sin\alpha &= \cos(\alpha - \varphi)\quad\textrm{where }\tan\varphi = \frac{B}{A}\\ \sin^2\theta &= \frac{1 - \cos(2\theta)}{2}\\ \cos^2(\theta) &= \frac{1 + \cos(2\theta)}{2}\\ \cos^3(\theta) &= \frac{3}{4}\cos\theta + \frac{1}{4}\cos 3\theta\\ \mathrm{E}[\sin^2(t)] &= \frac{1}{2}\\ \end{align*}\]