Precalculus — Formula Reference

Where the precalc story lies

The seams

Every ⚠ from the tables below, pulled up front as the spine — the places where the precalc-level statement papers over something rigor will later care about.

• §1.1 Exponent laws — ⚠ $0^0$ is context-dependent (1 in combinatorics/series; indeterminate as a limit form)
• §1.2 Radicals — ⚠ In particular $\sqrt{x^2} = \lvert x\rvert$, not $x$. First place naïve algebra and rigor diverge.
• §1.3 Logarithms — ⚠ log is only defined for positive arguments
• §1.3 Logarithms — ⚠ no rule for $\log_a(x+y)$
• §3.2 Absolute value & inequalities — ⚠ Multiplying or dividing an inequality by a negative number reverses its direction.
• §5.1 Function properties (inverse relations) — ⚠ $f^{-1} \neq 1/f$
• §5.2 Transformations (horizontal shift) — ⚠ sign is counterintuitive
• §8 Unit circle — ⚠ $\tan,\sec$ undefined where $\cos\theta = 0$
• §10.2 Sum & difference ($\cos(A \pm B)$) — ⚠ sign flips relative to the argument
• §12 Conic sections — ⚠ Note the sign convention difference: ellipse uses $c^2 = a^2 - b^2$, hyperbola uses $c^2 = a^2 + b^2$.
• §13 Sequences & series — ⚠ infinite geometric series diverges if $\lvert r\rvert \geq 1$

Full reference

*Lookup reference for settled computational material. Conditions are stated explicitly. ⚠ marks places where the precalc-level statement papers over something rigor will later care about.

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1. Algebra fundamentals: Exponents, Radicals, Logarithms

1.1 Exponent laws

Formula	Conditions	Note
$a^m a^n = a^{m+n}$	real $a$	combine like bases
$\dfrac{a^m}{a^n} = a^{m-n}$	$a \neq 0$
$(a^m)^n = a^{mn}$		power of a power
$(ab)^n = a^n b^n$		distributes over product
$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$b \neq 0$
$a^0 = 1$	$a \neq 0$	⚠ $0^0$ is context-dependent (1 in combinatorics/series; indeterminate as a limit form)
$a^{-n} = \dfrac{1}{a^n}$	$a \neq 0$
$a^{1/n} = \sqrt[n]{a}$	if $n$ even, $a \geq 0$	rational exponent = root
$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$	if $n$ even, $a \geq 0$

1.2 Radicals

Formula	Conditions	Note
$\sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}$	if $n$ even, $a,b \geq 0$
$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$	$b \neq 0$; if $n$ even, $a,b \geq 0$
$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$
$\sqrt[n]{a^n} = a$	$n$ odd
$\sqrt[n]{a^n} = \lvert a\rvert$	$n$ even	⚠ In particular $\sqrt{x^2} = \lvert x\rvert$, not $x$. First place naïve algebra and rigor diverge.

1.3 Logarithms

Throughout: $a > 0$, $a \neq 1$.

Formula	Conditions	Note
$y = \log_a x \iff a^y = x$	$x > 0$	⚠ log is only defined for positive arguments
$\log_a 1 = 0,\quad \log_a a = 1$
$\log_a(a^x) = x,\quad a^{\log_a x} = x$	second needs $x>0$	log and exp are inverses
$\log_a(xy) = \log_a x + \log_a y$	$x, y > 0$	⚠ no rule for $\log_a(x+y)$
$\log_a\!\left(\dfrac{x}{y}\right) = \log_a x - \log_a y$	$x, y > 0$
$\log_a(x^r) = r\log_a x$	$x > 0$, real $r$
$\log_a x = \dfrac{\log_b x}{\log_b a} = \dfrac{\ln x}{\ln a}$	usual domain	change of base

Notation: $\ln = \log_e$ (natural), $\log = \log_{10}$ (common, in Stewart).

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2. Special products & factoring

2.1 Special products

Formula	Note
$(a+b)^2 = a^2 + 2ab + b^2$
$(a-b)^2 = a^2 - 2ab + b^2$
$(a+b)(a-b) = a^2 - b^2$	difference of squares
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
$(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$

2.2 Factoring

Formula	Note
$a^2 - b^2 = (a-b)(a+b)$
$a^3 - b^3 = (a-b)(a^2+ab+b^2)$
$a^3 + b^3 = (a+b)(a^2-ab+b^2)$
$a^n - b^n = (a-b)(a^{n-1}+a^{n-2}b+\dots+b^{n-1})$	general

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3. Equations & inequalities

3.1 Quadratics

Formula	Conditions	Note
$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$	$a \neq 0$	quadratic formula
$D = b^2 - 4ac$		discriminant: $D>0$ two real roots, $D=0$ one double root, $D<0$ two complex conjugates
$x_1 + x_2 = -\dfrac{b}{a},\quad x_1 x_2 = \dfrac{c}{a}$		Vieta’s relations
$x^2 + bx = \left(x+\tfrac{b}{2}\right)^2 - \tfrac{b^2}{4}$		completing the square

3.2 Absolute value & inequalities

Statement	Equivalent to	Conditions
$\lvert x\rvert = a$	$x = \pm a$	$a \geq 0$
$\lvert x\rvert < a$	$-a < x < a$	$a > 0$
$\lvert x\rvert > a$	$x < -a$ or $x > a$	$a > 0$

⚠ Multiplying or dividing an inequality by a negative number reverses its direction.

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4. Coordinate geometry & lines

Formula	Conditions	Note
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$		distance between two points
$\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$		midpoint
$(x-h)^2 + (y-k)^2 = r^2$		circle, center $(h,k)$, radius $r$
$m = \dfrac{y_2-y_1}{x_2-x_1}$	$x_1 \neq x_2$	slope
$y - y_1 = m(x-x_1)$		point–slope
$y = mx + b$		slope–intercept
$Ax + By + C = 0$	$A,B$ not both 0	general form
$m_1 = m_2$		parallel lines
$m_1 m_2 = -1$	neither vertical	perpendicular lines

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5. Functions & transformations

5.1 Properties

Concept	Definition	Note
even function	$f(-x) = f(x)$	symmetric about $y$-axis
odd function	$f(-x) = -f(x)$	symmetric about origin
composition	$(f \circ g)(x) = f(g(x))$	need $g(x) \in \operatorname{dom} f$
inverse	$f^{-1}$ exists $\iff f$ one-to-one	horizontal line test
inverse relations	$f(f^{-1}(x)) = x,\ f^{-1}(f(x)) = x$	graph reflects over $y=x$; ⚠ $f^{-1} \neq 1/f$

5.2 Transformations (take $c > 0$)

Transformation	Effect
$y = f(x) + c$ / $y = f(x) - c$	shift up / down by $c$
$y = f(x-c)$ / $y = f(x+c)$	shift right / left by $c$ — ⚠ sign is counterintuitive
$y = -f(x)$	reflect over $x$-axis
$y = f(-x)$	reflect over $y$-axis
$y = c\,f(x)$	vertical stretch ($c>1$) / shrink ($0<c<1$)
$y = f(cx)$	horizontal shrink ($c>1$) / stretch ($0<c<1$)

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6. Polynomial & rational functions

Formula / Result	Conditions	Note
$f(x) = a(x-h)^2 + k$		quadratic vertex form, vertex $(h,k)$
$h = -\dfrac{b}{2a},\quad k = f(h)$		vertex from standard form
Remainder theorem: $f(c)$ = remainder of $f(x) \div (x-c)$
Factor theorem: $(x-c)$ is a factor $\iff f(c) = 0$
Rational root theorem: candidate roots $\pm\tfrac{p}{q}$	$p \mid$ constant, $q \mid$ leading coeff	only a candidate list, not a guarantee
Fundamental theorem of algebra: a degree-$n$ polynomial has $n$ complex roots	counted with multiplicity

Rational function asymptotes ($\deg$ = degree of numerator $N$ vs denominator $D$):

Case	Horizontal / slant behavior
$\deg N < \deg D$	horizontal asymptote $y = 0$
$\deg N = \deg D$	horizontal asymptote $y = \tfrac{\text{lead } N}{\text{lead } D}$
$\deg N = \deg D + 1$	slant (oblique) asymptote

Vertical asymptotes occur at zeros of $D$ that do not cancel with $N$.

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7. Exponential & logarithmic functions

Formula	Conditions	Note
$f(x) = a^x$	$a>0,\ a\neq 1$	domain $\mathbb{R}$, range $(0,\infty)$; growth if $a>1$, decay if $0<a<1$
$e \approx 2.71828$		base of natural exponential
$A = P\left(1 + \tfrac{r}{n}\right)^{nt}$		compound interest, $n$ periods/year
$A = Pe^{rt}$		continuous compounding
$A(t) = A_0 e^{kt}$		exp. growth ($k>0$) / decay ($k<0$)
doubling time $= \tfrac{\ln 2}{k}$; half-life $= \tfrac{\ln 2}{\lvert k\rvert}$

(Log laws: see §1.3.)

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8. Trigonometry — angles & the unit circle

Formula	Conditions	Note
$\pi \text{ rad} = 180^\circ$		degree ↔ radian
$s = r\theta$	$\theta$ in radians	arc length
$A = \tfrac{1}{2} r^2 \theta$	$\theta$ in radians	sector area
$\sin\theta = \tfrac{y}{r},\ \cos\theta = \tfrac{x}{r},\ \tan\theta = \tfrac{y}{x}$	$r = \sqrt{x^2+y^2}$	on unit circle $r=1$
$\csc\theta = \tfrac{1}{\sin\theta},\ \sec\theta = \tfrac{1}{\cos\theta},\ \cot\theta = \tfrac{1}{\tan\theta}$	denominators $\neq 0$	reciprocals
$\tan\theta = \tfrac{\sin\theta}{\cos\theta}$	$\cos\theta \neq 0$	⚠ $\tan,\sec$ undefined where $\cos\theta = 0$

Special values:

$\theta$	$0$	$\tfrac{\pi}{6}$	$\tfrac{\pi}{4}$	$\tfrac{\pi}{3}$	$\tfrac{\pi}{2}$
$\sin$	$0$	$\tfrac{1}{2}$	$\tfrac{\sqrt2}{2}$	$\tfrac{\sqrt3}{2}$	$1$
$\cos$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac{\sqrt2}{2}$	$\tfrac{1}{2}$	$0$
$\tan$	$0$	$\tfrac{\sqrt3}{3}$	$1$	$\sqrt3$	undef.

Periods: $\sin,\cos,\csc,\sec$ have period $2\pi$; $\tan,\cot$ have period $\pi$.

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9. Trigonometry — right triangles & solving triangles

Formula	Conditions	Note
$\sin\theta = \tfrac{\text{opp}}{\text{hyp}},\ \cos\theta = \tfrac{\text{adj}}{\text{hyp}},\ \tan\theta = \tfrac{\text{opp}}{\text{adj}}$	right triangle	SOH-CAH-TOA
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$		law of sines (watch the ambiguous SSA case)
$c^2 = a^2 + b^2 - 2ab\cos C$		law of cosines (and cyclic permutations)
Area $= \tfrac{1}{2}ab\sin C$		two sides + included angle
Area $= \sqrt{s(s-a)(s-b)(s-c)}$	$s = \tfrac{a+b+c}{2}$	Heron’s formula

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10. Trigonometric identities

10.1 Fundamental

Identity
$\sin^2\theta + \cos^2\theta = 1$
$1 + \tan^2\theta = \sec^2\theta$
$1 + \cot^2\theta = \csc^2\theta$
$\cos(-\theta) = \cos\theta$ (even)	$\sin(-\theta) = -\sin\theta,\ \tan(-\theta) = -\tan\theta$ (odd)
$\sin\!\left(\tfrac{\pi}{2}-\theta\right) = \cos\theta$	cofunction (similarly for the others)

10.2 Sum & difference

Identity
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ — ⚠ sign flips relative to the argument
$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

10.3 Double angle

Identity
$\sin 2\theta = 2\sin\theta\cos\theta$
$\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$
$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$

10.4 Half angle & power-reducing

Identity	Conditions
$\sin\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1-\cos\theta}{2}}$	sign by quadrant
$\cos\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1+\cos\theta}{2}}$	sign by quadrant
$\tan\tfrac{\theta}{2} = \dfrac{1-\cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1+\cos\theta}$
$\sin^2\theta = \dfrac{1-\cos 2\theta}{2},\quad \cos^2\theta = \dfrac{1+\cos 2\theta}{2}$	power-reducing

10.5 Product-to-sum & sum-to-product

Identity
$\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]$
$\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]$
$\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]$
$\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}$
$\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}$
$\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}$
$\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}$

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11. Polar, parametric, vectors, complex numbers

Formula	Conditions	Note
$x = r\cos\theta,\ y = r\sin\theta$		polar → rectangular
$r^2 = x^2 + y^2,\ \tan\theta = \tfrac{y}{x}$	$x \neq 0$	rectangular → polar (mind the quadrant)
$z = a + bi,\quad \lvert z\rvert = \sqrt{a^2+b^2}$		complex modulus
$z = r(\cos\theta + i\sin\theta)$		polar form
$z_1 z_2 = r_1 r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$		multiplication adds angles
$z^n = r^n(\cos n\theta + i\sin n\theta)$		De Moivre’s theorem
$z^{1/n} = r^{1/n}\!\left[\cos\tfrac{\theta + 2\pi k}{n} + i\sin\tfrac{\theta + 2\pi k}{n}\right]$	$k = 0,\dots,n-1$	$n$ distinct roots
$\lvert \mathbf{v}\rvert = \sqrt{a^2+b^2}$	$\mathbf{v} = \langle a,b\rangle$	vector magnitude
$\mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2 = \lvert\mathbf{u}\rvert\lvert\mathbf{v}\rvert\cos\theta$		dot product

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12. Conic sections

Conic	Standard form (centered at origin)	Key quantities
Parabola (vertical)	$x^2 = 4py$	focus $(0,p)$, directrix $y = -p$
Parabola (horizontal)	$y^2 = 4px$	focus $(p,0)$, directrix $x = -p$
Ellipse	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\ a>b$	$c^2 = a^2 - b^2$, foci $(\pm c, 0)$
Hyperbola	$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$	$c^2 = a^2 + b^2$, asymptotes $y = \pm\tfrac{b}{a}x$

Center $(h,k)$: replace $x \to x-h$, $y \to y-k$. Eccentricity $e = \tfrac{c}{a}$. ⚠ Note the sign convention difference: ellipse uses $c^2 = a^2 - b^2$, hyperbola uses $c^2 = a^2 + b^2$.

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13. Sequences & series

Formula	Conditions	Note
$a_n = a_1 + (n-1)d$		arithmetic, common difference $d$
$S_n = \tfrac{n}{2}(a_1 + a_n) = \tfrac{n}{2}[2a_1 + (n-1)d]$		arithmetic partial sum
$a_n = a_1 r^{\,n-1}$		geometric, common ratio $r$
$S_n = a_1\dfrac{1 - r^n}{1 - r}$	$r \neq 1$	geometric partial sum
$S = \dfrac{a_1}{1 - r}$	$\lvert r\rvert < 1$	⚠ infinite geometric series diverges if $\lvert r\rvert \geq 1$
$\sum_{k=1}^{n} k = \tfrac{n(n+1)}{2}$
$\sum_{k=1}^{n} k^2 = \tfrac{n(n+1)(2n+1)}{6}$
$\sum_{k=1}^{n} k^3 = \left[\tfrac{n(n+1)}{2}\right]^2$

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14. Binomial theorem & counting

Formula	Conditions	Note
$(a+b)^n = \sum_{k=0}^{n}\binom{n}{k} a^{n-k} b^k$	$n$ a non-negative integer	binomial theorem
$\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$	$0 \le k \le n$	combinations
$P(n,k) = \dfrac{n!}{(n-k)!}$		permutations (order matters)
$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$		Pascal’s identity