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Combinatorics I
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Algebra · 5Calculus · 2Probability · 5Pure math · 4Trigonometry · 4
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ACT Trigonometry
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Binomial Distribution
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Expected Value & Linearity
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Hypergeometric Distribution
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The Matrix Trace
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Lesson library
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Flashcards strengthen recall. These lessons build the reasoning, examples, and problem-solving methods behind each formula.
PDF
Introductory Statistics
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A comprehensive introductory statistics course from OpenStax, covering descriptive statistics, probability, discrete and continuous random variables, normal distributions, and more.
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Linear Algebra
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An open-source textbook on linear algebra, focusing on systems of linear equations, vector spaces, linear maps, determinants, and eigenvalues.
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Algebra Primer
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A specially curated native introductory guide covering core algebraic concepts.
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Combinatorics I
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An introduction to combinations, permutations, and discrete counting principles.
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Counting Specific Poker Hands
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A focused lesson on the Rank-Then-Suit strategy for counting exact poker hand configurations using combinations.
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Burnside's Lemma
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A step-by-step guide to Burnside's Lemma for counting distinct objects under rotational symmetry.
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Probability Fundamentals
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A broad introduction to probability: sample spaces, events, axioms, and the classical formula. Covers complementary counting, uniform probability, and worked real-world examples.
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Binomial Distribution
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The binomial model for repeated independent trials. Covers the binomial formula, mean, variance, and normal approximation with worked examples.
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Conditional Probability & Bayes' Theorem
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How new information updates probability estimates. Covers conditional probability, the multiplication rule, and Bayes' theorem with medical-test and dice examples.
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Expected Value & Linearity
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The weighted average of a random variable's outcomes. Covers the definition, linearity of expectation, and applications to dice, games, and the coupon-collector problem.
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Geometric Distribution
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Counts the number of trials until the first success. Covers the PMF, CDF, memoryless property, and expected value.
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Poisson Distribution
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Models rare events over a fixed interval. Covers the Poisson PMF, relationship to the binomial, and the Poisson approximation theorem.
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Negative Binomial Distribution
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Counts trials until the r-th success. Covers the PMF, relationship to the geometric distribution, and worked examples.
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Hypergeometric Distribution
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Sampling without replacement from a finite population. Covers the PMF, comparison with binomial, and quality-control applications.
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Geometric Probability
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Probability defined by areas and lengths in continuous spaces. Covers uniform distributions on intervals and regions, and dart-board style examples.
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Independent Events
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When the occurrence of one event has no effect on another. Covers the product rule, mutual independence, and common misconceptions.
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The Monty Hall Problem
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A deep dive into the famous three-door paradox. Covers the conditional probability argument, simulation intuition, and generalizations.
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Martingales & the Gambler's Ruin
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Fair-game stochastic processes and stopping times. Covers martingale definition, optional stopping theorem, and the Gambler's Ruin problem.
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Pólya Urn & Exchangeability
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A reinforcement model for probability. Covers the Pólya urn process, exchangeability, and the Beta-Binomial connection.
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Probability Generating Functions
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A transform technique for discrete distributions. Covers the PGF definition, recovering probabilities and moments, and convolution of distributions.
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Random Walks
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Sequential coin-flip paths on the integer line. Covers simple random walks, return-to-origin probability, the reflection principle, and streak probabilities.
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The Birthday Problem
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How many people does it take before two share a birthday? Covers the complementary counting approach, the birthday paradox formula, and generalizations.
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Algebra Essentials
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Core algebraic techniques: solving quadratics, factoring, sequences, logarithms, and polynomial theorems. Includes the quadratic formula, Vieta's formulas, and the Factor Theorem.
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Number Theory
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Divisibility, primes, modular arithmetic, and number-theoretic functions. Covers GCD, LCM, Euler's totient, Wilson's theorem, and the Chinese Remainder Theorem.
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Derivatives of Trigonometric Functions
By MathMech Team
An introduction to the derivatives of sine, cosine, and tangent. Covers the geometric definition of trig derivatives, key formulas, and applications.
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The Unit Circle
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Exact values of sine, cosine, and tangent at the fundamental angles using the unit circle. Covers the construction of the unit circle and all quadrant signs.
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Trigonometric Ratios
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SOH-CAH-TOA and the definitions of the six trig functions via right-triangle ratios. Covers the 30-60-90 and 45-45-90 special triangles.
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ACT Trigonometry
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A focused ACT Math lesson covering right-triangle ratios, unit-circle signs, identities, interval equations, the Laws of Sines and Cosines, arc length, and sinusoidal models.
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Quadrant-Constrained Double-Angle Identities
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An exact-value trigonometry subtopic where quadrant information determines signs before a double-angle identity is applied.
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Reciprocal Trig Functions
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Definitions and values of cosecant, secant, and cotangent. Covers their relationship to sin, cos, tan, and exact values at standard angles.
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Pythagorean Identities
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The three fundamental Pythagorean identities and how to derive and apply them. Covers proofs from the unit circle and uses in simplifying expressions.
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Evaluating Trigonometric Expressions
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A step-by-step guide to evaluating expressions involving sec, sin, and tan at special angles. Covers reciprocal trig functions, Pythagorean identities, and the 30-60-90 triangle.
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Identity-First Exact Trig Expression Evaluation
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A focused lesson for expressions that pair secant-squared minus tangent-squared with squared cosecant at a special angle, using identity recognition before exact-value evaluation.
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Evaluating Trig Expressions (Practice 1)
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A detailed lesson on evaluating sine, cosine, and tangent expressions using the sine-cosine Pythagorean identity.
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Evaluating Trig Expressions (Practice 2)
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A detailed lesson on evaluating sine, cosine, and tangent expressions using the fundamental Pythagorean identity and special right triangles.
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Evaluating Trig Expressions (Practice 3)
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A detailed lesson on evaluating cosecant and cotangent expressions using the cosecant-cotangent Pythagorean identity.
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Complementary Sine and Cosine Values
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Tangent and Cotangent Reciprocal Sum
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Double-Angle Sine at 45 Degrees
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Double-Angle Cosine at 30 Degrees
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Secant–Tangent Pythagorean Identity
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Cosecant–Cotangent Pythagorean Identity
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Sine–Cosine Pythagorean Identity
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Tangent Squared at 60 Degrees
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Cosecant Minus Cotangent at 30 Degrees
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Sine Double Angle at 30 Degrees
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Cosine of a Sum of Angles
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Sine with a 150-Degree Reference Angle
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Cosine in Quadrant II at 135 Degrees
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Tangent as a Sine–Cosine Ratio
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Secant as the Reciprocal of Cosine
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A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
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Matrix Addition & Scalar Multiplication
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Element-wise matrix operations. Covers matrix addition, subtraction, scalar multiplication, and the zero matrix.
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The Matrix Trace
By MathMech Team
The sum of the diagonal entries of a square matrix. Covers the definition, linearity, cyclic property, and relationship to eigenvalues.
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The Determinant (2×2)
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Computing and interpreting the determinant of a 2×2 matrix. Covers the ad-bc formula, geometric meaning as area scaling, and conditions for invertibility.
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Determinants (General)
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Determinants of n×n matrices via cofactor expansion. Covers properties, the identity matrix determinant, and the effect of row operations.
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Matrix Properties & Powers
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Key algebraic properties of matrices including the determinant of a matrix power, transpose rules, and invertibility conditions.
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The Dot Product
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The inner product of two vectors. Covers the algebraic and geometric definitions, projection, angle between vectors, and the Cauchy-Schwarz inequality.
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Orthogonal Vectors
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Vectors that meet at a right angle. Covers the dot-product test for orthogonality, orthogonal complements, and the Gram-Schmidt process.
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Singular Matrices & Linear Dependence
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When a matrix has no inverse. Covers linear dependence of columns, zero determinant, and the connection to null spaces and inconsistent systems.
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Visual Regression Tests
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Quality assurance test book for verifying layout, styles, and KaTeX math equation rendering.
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