Formulas & Methods

Study Sheet

Formulas & Methods

The AMC 12 formula sheet: trig, logs, complex numbers, and beyond

Trigonometry & Complex Numbers

Tip
Identities

sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1; sin⁡(a±b)\sin(a \pm b), cos⁡(a±b)\cos(a \pm b), tan⁡(a±b)\tan(a \pm b); double angle sin⁡2a=2sin⁡acos⁡a\sin 2a = 2\sin a\cos a, cos⁡2a=2cos⁡2a−1=1−2sin⁡2a\cos 2a = 2\cos^2a - 1 = 1 - 2\sin^2a; half angle; product-to-sum 2sin⁡acos⁡b=sin⁡(a+b)+sin⁡(a−b)2\sin a\cos b = \sin(a+b) + \sin(a-b); sum-to-product. RR-form: asin⁡x+bcos⁡x=a2+b2sin⁡(x+ϕ)a\sin x + b\cos x = \sqrt{a^2+b^2}\sin(x + \phi).

Convert everything to sines and cosines of one angle, or to a single trig function via the RR-form. Exact values: 15∘15^\circ, 18∘18^\circ, 36∘36^\circ, 75∘75^\circ from half/difference formulas and the golden ratio.

Tip
Triangles

Law of sines asin⁡A=2R\tfrac{a}{\sin A} = 2R; law of cosines; area 12absin⁡C\tfrac12ab\sin C; Area=abc4R=rs\text{Area} = \tfrac{abc}{4R} = rs. Stewart, Apollonius (median), angle bisector length bc(1−a2(b+c)2)\sqrt{bc\left(1 - \tfrac{a^2}{(b+c)^2}\right)}.

Reminder — Law of Sines:  asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
Reminder — Law of Cosines:  c2=a2+b2−2abcos⁡Cc^{2}=a^{2}+b^{2}-2ab\cos C

Given SSS use cosines; ASA/AAS use sines; SSA check for the ambiguous case. Two area formulas equated solve for the unknown.

Tip
Complex numbers

∣z∣2=zzˉ|z|^2 = z\bar z; z=reiθz = re^{i\theta}; de Moivre (reiθ)n=rneinθ(re^{i\theta})^n = r^ne^{in\theta}; nnth roots of unity sum to 00 and are equally spaced; xn−1=∏(x−ωk)x^n - 1 = \prod(x - \omega^k); ∣z−a∣|z - a| is a distance; multiplication rotates by arg⁡\arg.

Convert to polar for powers and roots. Use ω3=1\omega^3 = 1, 1+ω+ω2=01 + \omega + \omega^2 = 0 to collapse sums. Treat geometry problems on regular polygons as roots-of-unity problems.

Algebra & Polynomials

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Logs and exponentials

All log rules; log⁡ab⋅log⁡bc=log⁡ac\log_ab\cdot\log_bc = \log_ac; alog⁡ax=xa^{\log_ax} = x; xlog⁡by=ylog⁡bxx^{\log_by} = y^{\log_bx}. Exponential equations: match bases, or substitute u=axu = a^x.

Reminder — Logarithm rules:  log⁡b(xy)=log⁡bx+log⁡by,log⁡b ⁣xy=log⁡bx−log⁡by,log⁡b(xn)=nlog⁡bx\log_b(xy)=\log_b x+\log_b y,\quad \log_b\!\frac{x}{y}=\log_b x-\log_b y,\quad \log_b(x^{n})=n\log_b x

Take logs of products; exponentiate sums of logs. Check the domain (x>0x > 0 inside a log) after solving.

Tip
Polynomials in depth

Vieta for any degree; Newton's sums pk=e1pk−1−e2pk−2+⋯p_k = e_1p_{k-1} - e_2p_{k-2} + \cdots; remainder theorem P(a)P(a); factor theorem; conjugate root theorem; rational root theorem; roots-of-unity filter for coefficient sums; Descartes' rule for sign counts. Sum of coefficients =P(1)= P(1); alternating sum =P(−1)= P(-1).

Reminder — Remainder and Factor theorems:  P(x)=(x−c)Q(x)+P(c)⟹(x−c)∣P(x)  ⟺  P(c)=0P(x)=(x-c)Q(x)+P(c) \quad\Longrightarrow\quad (x-c)\mid P(x) \iff P(c)=0
Reminder — Vieta's formulas:  r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1 r_2=\frac{c}{a}

To evaluate symmetric expressions of roots, never find the roots. To count real roots, use derivatives (Rolle) and sign changes. Substitute x=1,−1,i,ωx = 1, -1, i, \omega to extract coefficient information.

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Sequences, series, recursion

Characteristic equation for linear recurrences; generating functions ∑anxn\sum a_nx^n turn recurrences into algebra; ∑nxn=x(1−x)2\sum nx^n = \tfrac{x}{(1-x)^2}; telescoping products; periodic recurrences (compute a cycle). Arithmetico-geometric sums by the shift-and-subtract trick.

Compute the first six terms of any recursion before theorizing. Look for a period. For sums with n⋅rnn\cdot r^n, differentiate the geometric series or subtract rSrS from SS.

Geometry, Counting & Probability

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Conics and coordinates

Circle (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2; parabola y2=4pxy^2 = 4px (focus (p,0)(p, 0)); ellipse x2a2+y2b2=1\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1, c2=a2−b2c^2 = a^2 - b^2; hyperbola c2=a2+b2c^2 = a^2 + b^2, asymptotes y=±baxy = \pm\tfrac bax. Distance from a point to a line; reflection across a line; rotation by 90∘90^\circ: (x,y)→(−y,x)(x, y) \to (-y, x).

Complete the square to identify a conic. Use the focus–directrix definition for "distance to a point equals distance to a line" problems.

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Advanced counting and probability

Inclusion–exclusion for nn sets; derangements Dn≈n!eD_n \approx \tfrac{n!}{e}; Catalan 1n+1(2nn)\tfrac{1}{n+1}\binom{2n}{n}; Vandermonde ∑(ak)(bn−k)=(a+bn)\sum\binom ak\binom{b}{n-k} = \binom{a+b}{n}; hockey stick ∑i=kn(ik)=(n+1k+1)\sum_{i=k}^n\binom ik = \binom{n+1}{k+1}. Geometric probability with areas; expected value by indicators; first-step equations for expected waiting times.

Translate "no two adjacent" into gaps, "paths not crossing" into Catalan, "sum of products" into Vandermonde. For expected values, write the linearity sum before anything else.

Concept
The universal problem-solving loop

Read twice. Restate the goal in your own words. List the givens. Pick a representation. Try small cases. Look for symmetry, an invariant, or an extremal object. Compute, then CHECK against a second method or a sanity bound.

At AMC 10/12 and AIME level, the second method is what separates a 44-second guess from a 1212-second certainty: compute a probability two ways (complement and direct), a length via two theorems, a count via a recurrence and a formula. Casework must be organized by a stated criterion so nothing is double-counted; algebra must be checked by substituting back.