Trigonometric Identities ======================== Euler's Formula --------------- \begin{align} e^{i\theta} = \cos \theta + i\sin \theta \end{align} Ptolemy’s Identities Derived from Euler's Formula ------------------------------------------------- ### Sum Formulae \begin{align} e^{i(\alpha + \beta)} &= e^{i\alpha} \cdot e^{i\beta} \\ \cos(\alpha + \beta) + i\sin(\alpha + \beta) &= (\cos \alpha + i\sin \alpha)(\cos \beta + i\sin \beta) \\ &= \cos\alpha\cos\beta - \sin\alpha\sin\beta + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta) \\ \cos(\alpha + \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta \end{align} ### Difference Formulae \begin{align} e^{i(\alpha - \beta)} &= e^{i\alpha} \cdot e^{-i\beta} \\ \cos(\alpha - \beta) + i\sin(\alpha - \beta) &= (\cos \alpha + i\sin \alpha)(\cos \beta - i\sin \beta) \\ &= \cos\alpha\cos\beta+\sin\alpha\sin\beta + i(\sin\alpha\cos\beta-\cos\alpha\sin\beta) \\ \cos(\alpha - \beta) &= \cos\alpha\cos\beta + \sin\alpha\sin\beta \\ \sin(\alpha - \beta) &= \sin\alpha\cos\beta - \cos\alpha\sin\beta \end{align} ### Product Identities \begin{align} \cos(\alpha + \beta) + \cos(\alpha - \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta + \cos\alpha\cos\beta + \sin\alpha\sin\beta \\ &= 2\cos\alpha\cos\beta \\ \cos\alpha\cos\beta &= \frac{\cos(\alpha + \beta) + \cos(\alpha - \beta)}{2} \\ \cos(\alpha + \beta) - \cos(\alpha - \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta - \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ &= -2\sin\alpha\sin\beta \\ \sin\alpha\sin\beta &= \frac{\cos(\alpha - \beta) - \cos(\alpha + \beta)}{2} \\ \sin(\alpha + \beta) + \sin(\alpha - \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta + \sin\alpha\cos\beta - \cos\alpha\sin\beta \\ &= 2\sin\alpha\cos\beta \\ \sin\alpha\cos\beta &= \frac{\sin(\alpha + \beta) + \sin(\alpha - \beta)}{2} \\ \sin(\alpha + \beta) - \sin(\alpha - \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta - \sin\alpha\cos\beta + \cos\alpha\sin\beta \\ &= 2\cos\alpha\sin\beta \\ \cos\alpha\sin\beta &= \frac{\sin(\alpha + \beta) - \sin(\alpha - \beta)}{2} \end{align} Double Angle Formulae Derived from Euler's Formula -------------------------------------------------- \begin{align} e^{i2\theta} &= e^{i\theta} \cdot e^{i\theta} \\ \cos(2\theta) + i\sin(2\theta) &= (\cos \theta + i\sin \theta)(\cos \theta + i\sin \theta) \\ &= \cos^2\theta - \sin^2\theta + i2\sin\theta\cos\theta \\ \cos(2\theta) &= \cos^2\theta - \sin^2\theta \\ \sin(2\theta) &= 2\sin\theta\cos\theta \end{align}