Formulæ (Group C)

N.B. The pupil need not commit to memory the formulæ marked thus “[”; but he should be able to work them out readily.

Formula connecting `E`, `F`	$E : F :: 9 : 10$
Formula connecting `E`, $Θ$	$E : Θ :: 180 : π$
Approximative values of `π`	$\frac{22}{7}$ , $\frac{355}{133}$ , 3⋅14159&c.
Reciprocal ratios	sin, cosec; cos, sec; tan, cot
Formula connecting sin, cos	${sin}^{2} + {cos}^{2} = 1$
” tan, sin, cos	$tan = \frac{sin}{cos}$
sec, tan	${sec}^{2} = {tan}^{2} + 1$
Sin, cos, tan, of 0°	0, 1, 0
90°	1, 0, $\frac{1}{0}$
180°	0, −1, 0
[270°	−1, 0, $\frac{1}{0}$
45°	$\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$ , 1
60°	$\frac{\sqrt{3}}{2}$ , $\frac{1}{2}$ , $\sqrt{3}$
30°	$\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ , $\frac{1}{\sqrt{3}}$
$Sin (A + B)$	$sin A . cos B + cos A . sin B$
” $(A - B)$	”−”
$Cos (A + B)$	$cos A . cos B - sin A . sin B$
” $(A - B)$	”+”
$Tan (A + B)$	$\frac{tan A + tan B}{1 - tan A . tan B}$
” $(A - B)$	$\frac{tan A - tan B}{1 + tan A . tan B}$
$Sin 2 A$	$2 . sin A . cos A$
$Cos 2 A$
in terms of cos, sin	${cos}^{2} A - {sin}^{2} A$
of cos only	$2 {cos}^{2} A - 1$
of sin only	$1 - 2 {sin}^{2} A$
$Tan 2 A$	$\frac{2 tan A}{1 - {tan}^{2} A}$
[ $Cos \frac{A}{2}$ , in terms of $cos A$	$\sqrt{\frac{1 + cos A}{2}}$
[ $Sin \frac{A}{2}$ , ”	$\sqrt{\frac{1 - cos A}{2}}$
${Tan}^{- 1} t_{1} + {tan}^{- 1} t_{2}$	${tan}^{- 1} \frac{t_{1} + t_{2}}{1 - t_{1} t_{2}}$
” − ”	${tan}^{- 1} \frac{t_{1} - t_{2}}{1 + t_{1} t_{2}}$
[Hence $2 {tan}^{- 1} t$	${tan}^{- 1} \frac{2 t}{1 - t^{2}}$
$Sin A + sin B$	$2 sin \frac{A + B}{2} . cos \frac{A - B}{2}$
” − ”	$2 cos \frac{A + B}{2} . sin \frac{A - B}{2}$
$Cos A + cos B$	$2 cos \frac{A + B}{2} . cos \frac{A - B}{2}$
” − ”	$- 2 sin \frac{A + B}{2} . sin \frac{A - B}{2}$

Triangles

Formulæ of sines	$\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$
” sides	$cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$
” tangents	$tan \frac{B - C}{2} = \frac{b - c}{b + c} . cot \frac{A}{2}$
$Cos \frac{A}{2}$ , in terms of sides	$\sqrt{\frac{s . (s - a)}{b c}}$
$sin \frac{A}{2}$ , ”	$\sqrt{\frac{(s - b) . (s - c)}{b c}}$
$tan \frac{A}{2}$ , ”	$\sqrt{\frac{(s - b) . (s - c)}{s . (s - a)}}$
[ $sin A$ , ”	$\frac{2}{b c} \sqrt{s . (s - a) . (s - b) . (s - c)}$
`a`, in terms of `b`, `c`, `B`, `C`	$b . cos C + c . cos B$

Area, in terms of two sides and included angle	$\frac{b c}{2} . sin A$
in terms of sides	$\sqrt{s . (s - a) . (s - b) . (s - c)}$

If Area be denoted by ‘`M`,’ and radii of inscribed, circumscribed, and escribed circles by ‘`r`, `R`, $R_{a}$ , $R_{b}$ , $R_{c}$ ;
`r`	$\frac{M}{s}$
`R`	$\frac{a b c}{4 M}$
[ $R_{a}$ , &c.	$\frac{M}{s - a}$ , &c.

Polygons. (`n` sides)

[Each angle	$180° - \frac{360°}{n}$
[Formula connecting `r`, `a`	$\frac{a}{2 r} = tan \frac{180°}{n}$
[”” `R`, `a`	$\frac{a}{2 R} = sin \frac{180°}{n}$
[Area, in terms of sides	$\frac{n a^{2}}{4} . cot \frac{180°}{n}$
[”” of `r`	$n r^{2} . tan \frac{180°}{n}$
[”” of `R`	$\frac{n R^{2}}{2} . sin \frac{360°}{n}$

Logarithms

If base be denoted by ‘`a`’;
$log a$	1
$log 1$	0
$log m n$	$log m + log n$
$log \frac{m}{n}$	$log m - log n$
$log m^{n}$	$n . log m$
$log \sqrt[n]{m}$	$\frac{log m}{n}$

Triangles

Polygons. (n sides)

Logarithms

Polygons. (`n` sides)