Grade 12 Mathematics Formula Sheet

α Algebra & Equations

Quadratic formula $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

Discriminant $\Delta = b^2 - 4ac$

$\Delta > 0$ 2 distinct real roots

$\Delta = 0$ 1 repeated real root

$\Delta < 0$ no real roots

Exponent laws $a^m \cdot a^n = a^{m+n}$

$\dfrac{a^m}{a^n} = a^{m-n}$

$(a^m)^n = a^{mn}$

$a^{-n} = \dfrac{1}{a^n}$ $\sqrt[n]{a} = a^{1/n}$

Log laws $\log(xy) = \log x + \log y$

$\log\!\left(\dfrac{x}{y}\right) = \log x - \log y$

$\log(x^n) = n\log x$

Change of base $\log_a x = \dfrac{\log x}{\log a}$

∑ Sequences & Series

Arith. term $T_n = a + (n-1)d$

Arith. sum $S_n = \dfrac{n}{2}\bigl[2a + (n-1)d\bigr]$

$S_n = \dfrac{n}{2}(a + l)$ [$l$ = last term]

Geom. term $T_n = a \cdot r^{\,n-1}$

Geom. sum $S_n = \dfrac{a(r^n - 1)}{r - 1}$ $r \neq 1$

Infinite sum $S_\infty = \dfrac{a}{1-r}$ $|r| < 1$

$a$ = first term $d$ = common difference $r$ = common ratio $n$ = number of terms

$ Finance, Growth & Decay

Simple interest $A = P(1 + in)$

Simple decay $A = P(1 - in)$

Compound growth $A = P(1 + i)^n$

Compound decay $A = P(1 - i)^n$

Effective rate $(1+i_{\text{eff}}) = \!\left(1 + \dfrac{i_{\text{nom}}}{m}\right)^{\!m}$

Future value $F = \dfrac{x\bigl[(1+i)^n - 1\bigr]}{i}$

Present value $P = \dfrac{x\bigl[1 - (1+i)^{-n}\bigr]}{i}$

$P$ = principal $A$ = accrued amount $i$ = rate per period $n$ = periods $x$ = payment $m$ = compoundings/year

∫ Differential Calculus

First principles $f'(x) = \displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$

Power rule $\dfrac{d}{dx}\bigl[x^n\bigr] = nx^{n-1}$

Constant $\dfrac{d}{dx}\bigl[k\bigr] = 0$

Scalar multiple $\dfrac{d}{dx}\bigl[k\,f(x)\bigr] = k\,f'(x)$

Sum / difference $\dfrac{d}{dx}\bigl[f \pm g\bigr] = f'(x) \pm g'(x)$

$f'(x) = 0$ stationary point (max or min)

$f''(x) = 0$ point of inflection

$f'(x) > 0$ function increasing

$f'(x) < 0$ function decreasing

$f''(x) > 0$ concave up (local min)

$f''(x) < 0$ concave down (local max)

θ Trigonometry

SOH-CAH-TOA $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$ $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$ $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

Pythag. identity $\sin^2\theta + \cos^2\theta = 1$

Quotient identity $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

Co-functions (90°) $\sin(90°\!-\!\theta)=\cos\theta$ $\cos(90°\!-\!\theta)=\sin\theta$

Reduction (180°) $\sin(180°\!-\!\theta)=\sin\theta$ $\cos(180°\!-\!\theta)=-\cos\theta$

$\sin(180°\!+\!\theta)=-\sin\theta$ $\cos(180°\!+\!\theta)=-\cos\theta$

Reduction (360°) $\sin(360°\!-\!\theta)=-\sin\theta$ $\cos(360°\!-\!\theta)=\cos\theta$

Negative angles $\sin(-\theta)=-\sin\theta$ $\cos(-\theta)=\cos\theta$

$\sin(\alpha \pm \beta)$ $\sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

$\cos(\alpha \pm \beta)$ $\cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

$\sin 2\alpha$ $2\sin\alpha\cos\alpha$

$\cos 2\alpha$ $\cos^2\!\alpha - \sin^2\!\alpha$

$1 - 2\sin^2\!\alpha$

$2\cos^2\!\alpha - 1$

Sine rule $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

Cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$

Area rule $\text{Area} = \tfrac{1}{2}ab\sin C$

	$0°$	$30°$	$45°$	$60°$	$90°$
$\sin\theta$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$	$0$
$\tan\theta$	$0$	$\dfrac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	undef

CAST rule — All positive (Q1) · Sin positive (Q2) · Tan positive (Q3) · Cos positive (Q4) | General solution adds $n\cdot360°$ (or $n\cdot180°$ for tan), $n\in\mathbb{Z}$

⊙ Analytical Geometry

Distance $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint $M = \!\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right)$

Gradient $m = \dfrac{y_2-y_1}{x_2-x_1} = \tan\theta$

Parallel $m_1 = m_2$

Perpendicular $m_1 \times m_2 = -1$

Line (slope-int.) $y = mx + c$

Line (point-slope) $y - y_1 = m(x - x_1)$

Circle at origin $x^2 + y^2 = r^2$

Circle at $(a,b)$ $(x-a)^2 + (y-b)^2 = r^2$

σ Statistics

Mean $\bar{x} = \dfrac{\sum x_i}{n}$

Variance $\sigma^2 = \dfrac{\sum(x_i - \bar{x})^2}{n}$

Std deviation $\sigma = \sqrt{\dfrac{\sum(x_i - \bar{x})^2}{n}}$

Regression line $\hat{y} = a + bx$

Slope $b$ $b = \dfrac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2}$

Intercept $a$ $a = \bar{y} - b\bar{x}$

Correlation: $r \approx +1$ strong positive · $r \approx -1$ strong negative · $r \approx 0$ no correlation

P Probability & Counting

Complement $P(A') = 1 - P(A)$

Addition rule $P(A\cup B) = P(A) + P(B) - P(A\cap B)$

Mutually excl. $P(A\cap B) = 0$

Independent $P(A\cap B) = P(A)\cdot P(B)$

Factorial $n! = n\times(n-1)\times\cdots\times 2\times 1$

Permutations ${}^{n}P_r = \dfrac{n!}{(n-r)!}$

Combinations ${}^{n}C_r = \dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

Counting princ. $n_1 \times n_2 \times \cdots \times n_k$

△ Euclidean Geometry — Key Circle Theorems

Chord & centre Perpendicular from centre bisects chord (and conversely)

Angle at centre Angle at centre $= 2\times$ angle at circumference (same arc)

Semicircle Angle in a semicircle $= 90°$

Cyclic quad. Opposite angles of a cyclic quad. are supplementary (sum $= 180°$)

Same chord Angles in the same segment (subtended by the same chord) are equal

Tangent ⊥ radius Tangent is perpendicular to radius at point of tangency

Tangent–chord Tangent–chord angle = inscribed angle in alternate segment

Equal tangents Two tangents from an external point are equal in length