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Trigonometry Cheat Sheet

Identities

Sum of angles:

\[\sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)\] \[\cos(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y)\]

In case $x = y$:

\[\sin(2x) = 2 \sin(x) \cos(x)\] \[\cos(2x) = \cos(x)^2 - \sin(x)^2\]

Exponentials

\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\] \[\sinh(z) = \frac{e^{z} - e^{-z}}{2}\] \[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\] \[\cosh(z) = \frac{e^z + e^{-z}}{2}\]

Hyperbolic

\[\sinh(x) = -i \sin(xi)\] \[\cosh(x) = \cos(xi)\] \[\cosh(x)^2 - \sinh(x)^2 = 1\]

Euler’s formula

For $x \in \mathbb{R}$:

\[e^{ix} = \cos(x) + i \sin(x)\]