Mathematics

Trigonometry and calculus

Trig ratios, identities, derivatives, integrals.

Basics

Trig ratios

Right triangle: sin=opp/hyp, cos=adj/hyp, tan=opp/adj. On the unit circle they are coordinates. A full turn is 2π radians. Physics and calculus default to radians.

Derivative

Instantaneous rate of change; slope of the tangent. Sum, product, quotient, chain rules. Extrema: f′=0 or undefined. The sign of f″ speaks to concavity.

Integral

A definite integral is signed area (limit of Riemann sums). Fundamental theorem: integrating the derivative recovers the function up to a constant. Substitution and parts invert the chain and product rules.

Formulas

Pythagorean identity

sin²θ + cos²θ = 1

Unit circle. 1+tan²θ=sec²θ is the same idea.

Angle-addition

sin(a±b) = sin a cos b ± cos a sin b

cos(a±b)=cos a cos b ∓ sin a sin b.

Law of sines

a / sin A = b / sin B = c / sin C = 2R

Any triangle. R is the circumradius.

Law of cosines

c² = a² + b² − 2 a b cos C

If C=90°, Pythagoras. Find an angle from three sides.

Definition of the derivative

f'(x) = lim_{h→0} [f(x+h) − f(x)] / h

If the limit exists, f is differentiable there.

Power rule

d/dx (xⁿ) = n xⁿ⁻¹

Workhorse for polynomials. d/dx eˣ = eˣ, d/dx ln x = 1/x.

Fundamental theorem

∫_a^b f′(x) dx = f(b) − f(a)

The sum of little changes is the net change.

Taylor (sketch)

f(x) ≈ f(a) + f′(a)(x−a) + f″(a)(x−a)² / 2! + ⋯

Approximate a smooth f by polynomials. Source of series for eˣ, sin, cos.

In this field

Algebra and geometry

Equations, exponentials and logs, Pythagoras, circles.

Statistics and linear algebra

Mean, variance, Bayes, matrices, eigenvalues.

Discrete, complex, series

Sequences, series, complex numbers, combinatorics.

Limits and number-theory sketch

Limits, continuity, primes, congruences.

Differential equations and series

First-order linear, harmonic oscillator, Taylor and geometric series.

Vectors and space

Vectors, dot and cross products, planes, and distance in 3D.

Conic sections

Parabola, ellipse, hyperbola, eccentricity, foci, and an orbit sketch.

Fourier sketch

Sines that rebuild a periodic function, coefficients, spectra, and a discrete-transform note.

Probability, continued

Expected value, variance, normal density, and a CLT one-liner.

Graph-theory sketch

Vertices and edges, degree, paths, Euler, and the adjacency matrix.