Basic Math.


Cramer’s rule

Given a system described by linear equations,

(1)   \begin{equation*} \begin{matrix} ax + by + cz &= {\color{orange}j}\\ dx + ey + fz &= {\color{orange}k}\\ gx + hy + iz&= {\color{orange}l} \end{matrix} \end{equation*}

and represented in matrix form as

(2)   \begin{equation*} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} {\color{red}j} \\ {\color{red}k} \\ {\color{red}l} \end{bmatrix} \end{equation*}

then, by Cramer’s rule, the values of x, y and z are given by

(3)   \begin{equation*} x = \frac { \begin{vmatrix} {\color{red}j} & b & c \\ {\color{red}k} & e & f \\ {\color{red}l} & h & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }, \quad y = \frac { \begin{vmatrix} a & {\color{red}j} & c \\ d & {\color{red}k} & f \\ g & {\color{red}l} & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} },\text{ and }z = \frac { \begin{vmatrix} a & b & {\color{red}j} \\ d & e & {\color{red}k} \\ g & h & {\color{red}l} \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} } \end{equation*}

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Trigonometric Identities

\sin(u \pm v) = \sin u ~ \cos v \pm \cos u ~ \sin v
\cos(u \pm v) = \cos u ~ \cos v \mp \sin u ~ \sin v
\tan(u \pm v) = {\tan u \pm \tan v \over 1 \mp \tan u ~ \tan v}