### 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: $a^{2}=b^{2}+c^{2}-2b c\cos A$ ### 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 ($x_{3}$, $y_{3}$): $A = \frac{1}{2} \det \begin{bmatrix}x_{1} & x_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{bmatrix}$ $\begin{array}{|l|l|l|}{\mathbf{Function:f(x)}}&{\mathbf{Definition}}&{\mathbf{Derivative}\mathbf{:}{\frac{d}{d x}}\bigl[f(x)\bigr]}\\ {\sinh(x)}&{{\frac{e^{x}-e^{-x}}{2}}}&{\cosh(x)}\\ {\cosh(x)}&{{\frac{e^{x}+e^{-x}}{2}}}&{\sinh(x)}\\ {\operatorname{tanh}(x)}&{{\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}}}&{1\!-\!\operatorname{tanh}^{2}(x)}\end{array}$