### Euler's Formula
$
e^{ix}=\cos x+i\sin x
$
#### Roots of Unity
The $k$ solutions to $z^{n}=1$ are:
$z_{k} = e^{2\pi i \cdot k/n}=\cos\left(2\pi\frac{k}{n} \right)+i\sin\left(2\pi\frac{k}{n} \right)$
for $k = 0, 1, 2, \dots n-1$
On the complex plane, the roots of unity are evenly spaced points on the unit circle.
### Law of sines
$
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
$
where A is the angle opposite of side $a$
### Law of cosines:
$a^{2}=b^{2}+c^{2}-2b c\cos A$
### Cosine formula
$\cos\theta = \frac{v\cdot w}{\lvert \lvert v \rvert \rvert ~\lvert \lvert w \rvert \rvert}$
### Cauchy-Schwartz Inequality
$\lvert v \cdot w \rvert \leq \lvert \lvert v \rvert \rvert ~ \lvert \lvert w \rvert \rvert$
### Triangle Inequality
$\lvert \lvert v + w \rvert \rvert \leq \lvert \lvert v \rvert \rvert + \lvert \lvert w \rvert \rvert$
### Other trig identities
$\begin{array}{l}{\sin(2x)=2\sin(x)\cos(x)}\\ {\cos(2x)=\cos^{2}(x)-\sin^{2}(x)}\\ {\tan(2x)={\frac{2\tan(x)}{1-\tan^{2}(x)}}}\end{array}$
$\begin{array}{l}{\sin(a\pm b)=\sin(a)\cos(b)\pm\cos(a)\sin(b)}\\ {\cos(a\pm b)=\cos(a)\cos(b)\mp\sin(a)\sin(b)}\\ {\tan(a\pm b)={\frac{\tan(a)\pm\tan(b)}{1\mp\tan(a)\tan(b)}}}\end{array}$
### Triangle Area
$A = \frac{1}{2}ab\sin C$
$A = \sqrt{ s(s-a)(s-b)(s-c) }$ , where $s = \frac{1}{2}(a+b+c)$
Area of a triangle with corners ($x_{1}$, $y_{1}$), ($x_{2}$, $y_{2}$), and ($x_{3}$, $y_{3}$):
$A = \frac{1}{2} \det \begin{bmatrix}x_{1} & x_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{bmatrix}$
### Rotation Matrix
2D rotation matrix:
$R_{\theta}=\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$
### Trig Graphs
$
\begin{array}{l}
\csc x = \frac{1}{\sin x} & & \sec x=\frac{1}{\cos x} & & \cot x=\frac{1}{\tan x}
\end{array}
$
![[Pasted image 20261014075333.png|435]]
### Hyperbolic Functions
| Definition | Derivative |
| -------------------------------------------- | -------------------------------------- |
| $\sinh(x) = \frac{e^{x}-e^{-x}}{2}$ | $\frac{d}{dx}\sinh(x)=\cosh(x)
lt;br> |
| ${} \cosh(x)=\frac{e^{x}+e^{-x}}{2} {}$ | $\frac{d}{dx}\cosh(x)=\sinh(x)$ |
| $\tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$ | $\frac{d}{dx}\tanh(x)=1-\tanh ^{2}(x)$ |
![[Pasted image 20260516093316.png|283]]