📏 Complete Formula Bank

ℹ️Quick Reference Guide
Comprehensive collection of all essential precalculus formulas with usage examples and micro-examples. Perfect for AMC 10/12 preparation and quick problem-solving reference.

🗂️ Table of Contents

🔄 Functions & Transformations

💡🎯 Function Mastery
Functions and transformations are fundamental to precalculus and appear frequently in AMC contests!

Domain Restrictions

⚠️⚠️ Critical for AMC
Always check domain restrictions when working with functions - this is a common source of errors!
Function TypeRestrictionExampleKey Insight
Square roots$\sqrt{f(x)}$ requires $f(x) \geq 0$$\sqrt{x-2}$ needs $x \geq 2$Radicand must be non-negative
Logarithms$\log_a f(x)$ requires $f(x) > 0$ and $a > 0, a \neq 1$$\log(x+1)$ needs $x > -1$Argument must be positive
Fractions$\frac{f(x)}{g(x)}$ requires $g(x) \neq 0$$\frac{1}{x-3}$ needs $x \neq 3$Denominator cannot be zero

Example: Domain of $\sqrt{x-2} + \log(x+1)$ is $x \geq 2$ and $x > -1$ → $x \geq 2$

Function Composition

Key Rules:

  • $(f \circ g)(x) = f(g(x))$ (apply $g$ first, then $f$)
  • $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$ (reverse order for inverses)

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(x+1) = (x+1)^2$

Transformations

📝📐 Transformation Rules
Master these transformation patterns for graphing and function analysis!
TypeFormulaEffectKey Insight
Vertical shift$f(x) + k$Shifts up $k$ unitsAdd to output
Horizontal shift$f(x-h)$Shifts right $h$ unitsSubtract from input
Vertical scaling$af(x)$Scales by factor $a$Multiply output
Horizontal scaling$f(bx)$Scales by factor $\frac{1}{b}$Divide input

Example: $f(x) = 2(x-1)^2 + 3$ is $x^2$ shifted right 1, scaled by 2, shifted up 3

🎯 Polynomials & Rational Functions

Key Concepts: Vieta’s formulas, factor theorems, and common factorizations

Vieta’s Formulas (Quadratic)

📊 For $ax^2 + bx + c = 0$ with roots $r_1, r_2$:

  • Sum: $r_1 + r_2 = -\frac{b}{a}$
  • Product: $r_1 r_2 = \frac{c}{a}$

💡 Example: $x^2 - 5x + 6 = 0$ has roots with sum 5 and product 6

Vieta’s Formulas (Cubic)

📊 For $ax^3 + bx^2 + cx + d = 0$ with roots $r_1, r_2, r_3$:

  • Sum: $r_1 + r_2 + r_3 = -\frac{b}{a}$
  • Sum of products: $r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a}$
  • Product: $r_1 r_2 r_3 = -\frac{d}{a}$

💡 Example: $x^3 - 6x^2 + 11x - 6 = 0$ has roots with sum 6, sum of products 11, product 6

Remainder & Factor Theorems

  • Remainder: When $p(x)$ is divided by $(x-a)$, remainder is $p(a)$
  • Factor: $(x-a)$ is a factor of $p(x)$ if and only if $p(a) = 0$

💡 Example: Remainder when $x^3 - 2x + 1$ is divided by $(x-2)$ is $8 - 4 + 1 = 5$

Common Factorizations

PatternFormula
Difference of squares$a^2 - b^2 = (a-b)(a+b)$
Perfect squares$a^2 \pm 2ab + b^2 = (a \pm b)^2$
Sum/difference of cubes$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

💡 Example: $x^3 - 8 = (x-2)(x^2 + 2x + 4)$

🔢 Exponents & Logarithms

Key Concepts: Exponent laws, logarithm properties, and special values

Exponent Laws

RuleFormula
Product$a^x \cdot a^y = a^{x+y}$
Quotient$\frac{a^x}{a^y} = a^{x-y}$
Power$(a^x)^y = a^{xy}$
Power of product$(ab)^x = a^x b^x$
Power of quotient$\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$

💡 Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

Logarithm Laws

RuleFormula
Product$\log_a(xy) = \log_a x + \log_a y$
Quotient$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
Power$\log_a(x^y) = y \log_a x$
Change of base$\log_a x = \frac{\log_b x}{\log_b a}$

💡 Example: $\log_2 8 = \frac{\log 8}{\log 2} = \frac{0.9031}{0.3010} = 3$

Special Values

ExpressionValueCondition
$a^0$$1$$a \neq 0$
$a^{-x}$$\frac{1}{a^x}$$a \neq 0$
$\log_a 1$$0$$a > 0, a \neq 1$
$\log_a a$$1$$a > 0, a \neq 1$

💡 Example: $5^0 = 1$, $2^{-3} = \frac{1}{8}$, $\log_3 1 = 0$, $\log_5 5 = 1$

📐 Trigonometry

💡🎯 Trigonometry Mastery
Trigonometry is essential for AMC contests. Master the unit circle and identities for success!

Unit Circle Values

⚠️⚠️ Must Memorize
These key values appear in 80% of trigonometry problems. Commit them to memory!
AngleRadians$\sin$$\cos$$\tan$Memory Trick
$0°$$0$$0$$1$$0$Starting point
$30°$$\frac{\pi}{6}$$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{3}$Small angle, small sine
$45°$$\frac{\pi}{4}$$\frac{\sqrt{2}}{2}$$\frac{\sqrt{2}}{2}$$1$Equal sine and cosine
$60°$$\frac{\pi}{3}$$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\sqrt{3}$Large angle, large sine
$90°$$\frac{\pi}{2}$$1$$0$undefinedMaximum sine

Fundamental Identities

📝🔑 Core Identities
These identities form the foundation of all trigonometric manipulations!
IdentityFormula
Pythagorean$\sin^2\theta + \cos^2\theta = 1$
Quotient$\tan\theta = \frac{\sin\theta}{\cos\theta}$
Reciprocal$\csc\theta = \frac{1}{\sin\theta}$
$\sec\theta = \frac{1}{\cos\theta}$
$\cot\theta = \frac{1}{\tan\theta}$

Example: If $\sin\theta = \frac{3}{5}$, then $\cos\theta = \pm\frac{4}{5}$ and $\tan\theta = \pm\frac{3}{4}$

Angle Transformation Identities

📝🔄 Reference Table
How trigonometric functions change with angle transformations - essential for simplifying expressions!
Function$\theta$$90°-\theta$$180°-\theta$$180°+\theta$$-\theta$Key Pattern
$\sin$$\sin\theta$$\cos\theta$$\sin\theta$$-\sin\theta$$-\sin\theta$Sine is odd
$\cos$$\cos\theta$$\sin\theta$$-\cos\theta$$-\cos\theta$$\cos\theta$Cosine is even
$\tan$$\tan\theta$$\cot\theta$$-\tan\theta$$\tan\theta$$-\tan\theta$Tangent is odd
$\cot$$\cot\theta$$\tan\theta$$-\cot\theta$$\cot\theta$$-\cot\theta$Cotangent is odd

Example: $\sin(180°-\theta) = \sin\theta$, $\cos(90°-\theta) = \sin\theta$, $\tan(-\theta) = -\tan\theta$

Addition Formulas

💡➕ Sum and Difference Identities
Essential for angle addition problems and simplifying complex trigonometric expressions!
$$ \begin{aligned} \sin(A \pm B) &= \sin A \cos B \pm \cos A \sin B \cr \cos(A \pm B) &= \cos A \cos B \mp \sin A \sin B \cr \tan(A \pm B) &= \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \end{aligned} $$

Example: $\sin 75° = \sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30° = \frac{\sqrt{6} + \sqrt{2}}{4}$

Sum-to-Product Formulas

🔄 Conversion Formulas: Transform sums/differences into products for easier manipulation

$$ \begin{aligned} \sin A + \sin B &= 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \cr \sin A - \sin B &= 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \cr \cos A + \cos B &= 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \cr \cos A - \cos B &= -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \end{aligned} $$

💡 Example: $\sin 75° + \sin 15° = 2\sin\left(\frac{75°+15°}{2}\right)\cos\left(\frac{75°-15°}{2}\right) = 2\sin 45° \cos 30° = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$

Double Angle Formulas

⚠️2× Formulas
Express trigonometric functions of double angles in terms of single angles - very common in AMC problems!

$$ \begin{aligned} \sin(2\theta) &= 2\sin\theta\cos\theta \cr \cos(2\theta) &= \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \cr \tan(2\theta) &= \frac{2\tan\theta}{1 - \tan^2\theta} \end{aligned} $$

Example: If $\sin\theta = \frac{3}{5}$, then $\sin(2\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

Half Angle Formulas

½ Formulas: Express trigonometric functions of half angles in terms of full angles

$$ \begin{aligned} \sin\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1 - \cos\theta}{2}} \cr \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1 + \cos\theta}{2}} \cr \tan\left(\frac{\theta}{2}\right) &= \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} \end{aligned} $$

💡 Example: If $\cos\theta = \frac{3}{5}$, then $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \frac{3}{5}}{2}} = \pm\sqrt{\frac{4}{5}} = \pm\frac{2\sqrt{5}}{5}$

Laws of Sines & Cosines

🔺 Triangle Laws: Essential for solving triangles with given information

  • Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
  • Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$
  • Area: $A = \frac{1}{2}ab\sin C$

💡 Example: Triangle with sides 3, 4, 5 has area $A = \frac{1}{2} \cdot 3 \cdot 4 \cdot \sin 90° = 6$

🌀 Complex Numbers

💡🎯 Complex Number Mastery
Complex numbers are essential for AMC 12 and advanced AMC 10 problems. Master these representations and operations!

Polar Form

Complex Number Representations:

FormFormulaDescription
Rectangular$z = a + bi$Standard form
Polar$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$Polar/exponential form
Modulus$|z| = \sqrt{a^2 + b^2}$Magnitude of $z$
Argument$\arg(z) = \arctan\left(\frac{b}{a}\right)$Angle with positive $x$-axis

Example: $1 + i = \sqrt{2}e^{i\pi/4} = \sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})$

Complex Conjugate

📝🔄 Conjugate Properties
The complex conjugate is crucial for simplifying expressions and finding moduli!
PropertyFormulaExampleKey Insight
Rectangular conjugate$\overline{a + bi} = a - bi$$\overline{3 + 4i} = 3 - 4i$Flip sign of imaginary part
Polar conjugate$\overline{re^{i\theta}} = re^{-i\theta}$$\overline{2e^{i\pi/3}} = 2e^{-i\pi/3}$Change sign of angle
Product with conjugate$z \cdot \overline{z} = |z|^2$$(3+4i)(3-4i) = 25$Always gives real number
Sum with conjugate$z + \overline{z} = 2\text{Re}(z)$$(3+4i) + (3-4i) = 6$Twice the real part
Difference with conjugate$z - \overline{z} = 2i\text{Im}(z)$$(3+4i) - (3-4i) = 8i$Twice imaginary part times $i$

Rotation Formulas

⚠️🔄 Geometric Transformations
Rotation formulas are powerful for geometric problems involving complex numbers!

Rotation Formulas:

TransformationFormula
Rotation by angle $\theta$$z^\prime = ze^{i\theta}$
Counterclockwise 90°$z^\prime = iz$
Counterclockwise 180°$z^\prime = -z$
Reflection across x-axis$z^\prime = \overline{z}$
TransformationFormulaExampleKey Insight
General rotation$z^{\prime} = ze^{i\theta}$Rotate $1+i$ by $\pi/2$: $(1+i)e^{i\pi/2} = (1+i)i = -1+i$Multiply by $e^{i\theta}$
90° rotation$z^{\prime} = iz$Rotate $2+3i$ by 90°: $i(2+3i) = -3+2i$Multiply by $i$
180° rotation$z^{\prime} = -z$Rotate $1+2i$ by 180°: $-(1+2i) = -1-2i$Multiply by $-1$

De Moivre’s Theorem

$$(re^{i\theta})^n = r^n e^{in\theta} = r^n(\cos(n\theta) + i\sin(n\theta))$$

Example: $(1 + i)^4 = (\sqrt{2}e^{i\pi/4})^4 = 4e^{i\pi} = 4(-1) = -4$

Roots of Unity

📝🎯 Roots of Unity
Roots of unity are essential for solving polynomial equations and geometric problems!
PropertyFormulaDescription
$n$-th roots$\omega_k = e^{i2\pi k/n}$$k = 0, 1, \ldots, n-1$
Sum$\sum_{k=0}^{n-1} \omega_k = 0$The sum of all $n$-th roots of unity
Product$\prod_{k=0}^{n-1} \omega_k = (-1)^{n-1}$The product of all $n$-th roots of unity
PropertyFormulaExampleKey Insight
$n$-th roots$\omega_k = e^{i2\pi k/n}$4th roots: $1, i, -1, -i$Equally spaced on unit circle
Sum of roots$\sum_{k=0}^{n-1} \omega_k = 0$$1 + i + (-1) + (-i) = 0$Always zero for $n > 1$
Product of roots$\prod_{k=0}^{n-1} \omega_k = (-1)^{n-1}$$1 \cdot i \cdot (-1) \cdot (-i) = 1$Depends on parity of $n$

📈 Sequences & Series

Key Concepts: Arithmetic and geometric sequences, binomial theorem

Arithmetic Sequences

  • General term: $a_n = a_1 + (n-1)d$
  • Sum: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$

Example: Sum of first 10 odd numbers: $S_{10} = \frac{10}{2}(1 + 19) = 100$

Geometric Sequences

  • General term: $a_n = a_1 r^{n-1}$
  • Finite sum: $S_n = a_1 \frac{1-r^n}{1-r}$ (for $r \neq 1$)
  • Infinite sum: $S_{\infty} = \frac{a_1}{1-r}$ (for $|r| < 1$)

Example: Sum of $1 + \frac{1}{2} + \frac{1}{4} + \cdots = \frac{1}{1-\frac{1}{2}} = 2$

Binomial Theorem

$$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$

Binomial coefficient: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Example: $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

📐 Coordinate Geometry

Key Concepts: Distance, lines, circles, and area formulas

Distance & Midpoint

  • Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
  • Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Example: Distance between $(1,2)$ and $(4,6)$ is $\sqrt{(4-1)^2 + (6-2)^2} = 5$

Lines

  • Slope: $m = \frac{y_2-y_1}{x_2-x_1}$
  • Point-slope: $y - y_1 = m(x - x_1)$
  • Slope-intercept: $y = mx + b$

Example: Line through $(1,2)$ with slope 3: $y - 2 = 3(x - 1)$ or $y = 3x - 1$

Circles

  • Standard form: $(x-h)^2 + (y-k)^2 = r^2$ (center $(h,k)$, radius $r$)
  • General form: $x^2 + y^2 + Dx + Ey + F = 0$

Example: Circle with center $(2,-3)$ and radius 5: $(x-2)^2 + (y+3)^2 = 25$

Area Formulas

  • Triangle: $A = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$
  • Shoelace formula: $A = \frac{1}{2}\left|\sum_{i=1}^n (x_i y_{i+1} - x_{i+1} y_i)\right|$

Example: Triangle with vertices $(0,0)$, $(3,0)$, $(0,4)$ has area $A = \frac{1}{2}|0(0-4) + 3(4-0) + 0(0-0)| = 6$

⚖️ Inequalities

Key Concepts: AM-GM, Cauchy-Schwarz, and triangle inequality

AM-GM Inequality

$\displaystyle \frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$
(for positive numbers) - **Equality**: When all numbers are equal

Example: $\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1$ → $x + \frac{1}{x} \geq 2$

Cauchy-Schwarz Inequality

  • $(\sum_{i=1}^n a_i b_i)^2 \leq (\sum_{i=1}^n a_i^2)(\sum_{i=1}^n b_i^2)$
  • Equality: When vectors are proportional

Example: $(2x + 3y)^2 \leq (2^2 + 3^2)(x^2 + y^2) = 13(x^2 + y^2)$

Triangle Inequality

  • $|a + b| \leq |a| + |b|$
  • $|a - b| \geq ||a| - |b||$

Example: $|x + y| \leq |x| + |y|$ for all real $x, y$

🎯 Vectors (AMC 12)

Key Concepts: Vector operations, dot product, and geometric applications

Basic Operations

  • Magnitude: $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$
  • Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$
  • Scalar multiplication: $k\vec{v} = \langle kv_1, kv_2 \rangle$

Example: $|\langle 3, 4 \rangle| = \sqrt{3^2 + 4^2} = 5$

Dot Product

  • Component form: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$
  • Geometric: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
  • Perpendicular: $\vec{a} \cdot \vec{b} = 0$

Example: $\langle 2, 3 \rangle \cdot \langle -1, 4 \rangle = 2(-1) + 3(4) = 10$

🎉🎉 You're Ready!
You now have a comprehensive precalculus formula reference! Practice regularly and use this as your go-to resource during contests.

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