MISC FORMULAS

$x_1+x_2=-\frac{B}{A}

$(x_1)(x_2)=(-1)^n \left(\frac{C}{A}\right)

$C=\frac{AB}{D+1}$
or
MODE TABLE
f(x)= (sum of variables' expo) C (x)
*n C r*

$A=\frac{1}{2}bh$
______________________________
$A=\frac{1}{2}\;ab\ sinC$
______________________________
$A=\frac{1}{2}a^2\ \frac{sinB\ sinC}{sinA}$
______________________________
$A=\sqrt{s(s-a)(s-b)(s-c)} $
______________________________

$y=mx+b

$m=\frac{y-y_1}{x-x_1}

$\log_bM=\frac{log\ M}{log\ b}

$\log_b{1}=0

$\log_b{b}=1

$\log_e{x}=lnx

$ln(e)=1

$vers(A)= \; 1-cos(A)

$covers(A)= \;1-sin(A)

$hav(A)=\dfrac {1}{2}(1-cos(A))$

$A_t=\dfrac{abc}{4R}$

Perpendicular Bisector of the Triangle that projects towards and MEETS at the
Center of the Circle

$A_t= rs$
Angle Bisector projects towards and meets at the Center of Circle

$\left(a+b\right)^n$
$r^{th}\ term\ =\left({_n}C_{r-1}\right)\;a^{n-r+1}\ \;b^{r-1}

$f\left(1,1,1,1,...\right)-f\left(0,0,0,0,...\right)\ $

$SoE =(SoE)_{ \ exponent\ of\ the\ variable\ of \ the\ given\ eqn}\,\,\,\, \left( _{n+r-1}C_r \right)

$\frac{n_e!}{\prod\left(n_{ti}!\right)}\left(\prod\left(K_{ei}\right)^{n_{ti}}\right)

$\theta=\ \frac{11\left(M\right)-\ 60\left(H\right)}{2}$

$F_n=\frac{1}{\sqrt5}\left[\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right]$

$S_n=\ \left(F_{n+2}\right)\ -\ \left(F_2\right)\

$L_n=\left(\frac{1+\sqrt5}{2}\right)^n+\left(\frac{1-\sqrt5}{2}\right)^n$

$S_n=\ \left(L_{n+2}\right)\ -\ \left(L_2\right)\

$L\left(f\left(t\right)\right)\ =\ \int_{0}^{\infty}{e^{-st}\ \ f\left(t\right)\ dt}$

Considering c is the LONGEST
____________________________________
$\downdownarrows RIGHT\ \Delta\downdownarrows\ $
$c^2=\ a^2\ +\ b^2$
____________________________________
$\downdownarrows OBTUSE\ \Delta\downdownarrows\ $
$c^2>\ a^2\ +\ b^2$
____________________________________
$\downdownarrows ACUTE\ \Delta\downdownarrows\ $
$c^2<\ a^2\ +\ b^2$

$MatA\ =\left|\begin{matrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\\\end{matrix}\right|$
$\Delta_{Area}=\ 0.5\ \ det\left(MatA\right)$

$y_{avg}=f_{avg}=\frac{1}{b-a}\int_{a}^{b}{f\left(x\right)\ dx}$

$ \#\ of\ DIAGS=\ \frac{S\left(S-3\right)}{2}$

$\int_{a}^{b}{\sqrt{1+\ \left(y'\right)^2}\ dx}$

$v=\ \frac{distance}{time}$

$d_{faster}-d_{slower}=\ 1\ circuit$

$d_{faster}+d_{slower}=\ 1\ circuit$

$Rate=\ \frac{Job\ Done}{Time}$

Combined Rate = Rate 1 ± Rate 2

$Job\ Done=\ \left(\#\ of\ MAN\right)\left(\#\ of\ DAYS\right)$
to finish the job
$\Sigma\ \left(Job\ Done\right)=\ total\ number\ of\ MAN-DAYS$

$x\ =\ \frac{60}{11}\ H\ $ or $x\ =\ \frac{2}{11}\ \theta$

$a_1\ ,\ {\ \ a}_2,\ {\ \ a}_3,{\ \ a}_4,\ {\ \ a}_5,\ {\ \ a}_6$
$d\ \ =\ {\ a}_2-a_1\ \ =\ a_3-a_2\ \ =a_4-a_3$

$a_n=a_1+\ \left(n-1\right)\ d$
or
$a_n=a_m+\ \left(n-m\right)\ d$

$S_n=\frac{n}{2}\ \left(a_1+a_n\right)$
____________________________
$S_n=\frac{n}{2}\ \left(2\ a_1+\left(n-1\right)\ d\right)$

$r=\ \frac{a_2}{a_1}\ \ =\ \frac{a_3}{a_2}\ =\ \frac{a_4}{a_3}\ =\ \frac{a_5}{a_4}$
$r=\ \frac{a_3-a_2}{a_2-a_1}$

$a_n=\ a_1\ r^{n-1}$
____________________________
$a_n=\ a_m\ r^{n-m}$

$S_n=\ \frac{a_1\ \left({1-r}^n\right)}{1-\ r}$

$S_n=\ \frac{a_1\ \left(r^n-1\right)}{r-\ 1}$

$for\ -1<r<1.0$
$S_n=\ \frac{a_1\ }{1-\ r}\ $

$d=\ \ \ \frac{1}{a_2}-\frac{1}{a_1}\ \ \ =\ \ \ \frac{1}{a_3}-\frac{1}{a_2}$

$a_n=\ \frac{1\ }{a_{1\ }+\ \left(n-1\right)\ d}\ $
or
$a_n=\ \frac{1\ }{a_{m\ }+\ \left(n-m\right)\ d}$

$S_n=\ \frac{1\ }{d}\ ln\ \left[\frac{2\ a_1+\ \left(2n-1\right)d}{2\ a_1-d}\right]$

$A.M.=\ \frac{x_1+\ x_2+x_3+x_n\ }{n}$

$G.M.=\ \sqrt[n\ \ ] {x_1\bullet\ x_2\bullet x_3\bullet x_n}$

$H.M.=\ \frac{n}{\frac{1}{x_1}+\ \frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_n}}$

$\left(A.M\right)\ \left(H.M\right)=\left(G.M\right)^2$

$n! = \sqrt{2\ \pi\ n}\ \left(\dfrac{n}{e}\right)^n$

$\frac{sin\ a}{sin\ A}\ =\ \frac{sin\ b}{sin\ B}\ =\ \frac{sin\ c}{sin\ C}$

$cos\ \left(a\right)=\ cos\ \left(b\right)\ cos\ \left(c\right)\ +\ sin\ \left(b\right)\ sin\ \left(c\right)\ cos\ \left(A\right)$
$cos\ \left(b\right)=\ cos\ \left(a\right)\ cos\ \left(c\right)\ +\ sin\ \left(a\right)\ sin\ \left(c\right)\ cos\ \left(B\right)$
$cos\ \left(c\right)=\ cos\ \left(a\right)\ cos\ \left(b\right)\ +\ sin\ \left(a\right)\ sin\ \left(b\right)\ cos\ \left(C\right)$

*notice the negative sign*
$cos\ \left(A\right)=\ -cos\ \left(B\right)\ cos\ \left(C\right)\ +\ sin\ \left(B\right)\ sin\ \left(C\right)\ cos\ \left(a\right)$
$cos\ \left(B\right)=\ -cos\ \left(A\right)\ cos\ \left(C\right)\ +\ sin\ \left(A\right)\ sin\ \left(C\right)\ cos\ \left(b\right)$
$cos\ \left(C\right)=\ -cos\ \left(A\right)\ cos\ \left(B\right)\ +\ sin\ \left(A\right)\ sin\ \left(B\right)\ cos\ \left(c\right)$

$\left(r\ \angle\ \theta\right)^n\ =\ r^n\ \angle\ \left(n\ \theta\right)$

TRIGO FUNCTIONS with $\pi$ (pi)
USE RAD MODE in CALCULATOR

$PRODUCTION\ \left(\frac{LCM}{hr}\right)$
$=\ C\bullet\ S\bullet\ V\bullet\ B\bullet\ E$

$PRODUCTION\ \left(\frac{LCM}{hr}\right)$
$=\ IO\bullet\ S\bullet\ E$