Usare formule

Skill — Usare formule.

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Linked atoms

254 linked atoms.

01 Numeri e operazioni fondamentali

• Properties of powers

02 Calcolo letterale

• Cube of a binomial
• Square of a binomial
• Sum times difference
• Tartaglia’s triangle
• The formulas for special products
• The method of shapes
• The triangle diagram for the square of a trinomial

04 Percentuali

• Absolute error and relative error
• Compound interest
• Discount and mark-up
• Percentage increase and decrease
• Periodic rate and fractional capitalisation
• Simple interest
• VAT and how to strip it out

06 Sistemi di equazioni lineari

• Cramer’s rule
• Inverse of a 2x2 matrix
• Row-by-column product
• Square and inverse of a 2x2 matrix
• The determinant of a 2x2 matrix

07 Teoria degli insiemi

• De Morgan’s laws

08 Fondamenti di geometria euclidea

• Sum of the interior angles of a polygon
• The intercept theorem
• The trapezium

10 Equazioni di secondo grado e fratte

• Cardano in action
• Cardano’s formula
• How a continued fraction is computed
• Sum and product of the roots
• The fractal star
• The quadratic formula and the discriminant
• The spiral of similar triangles
• Useful identities with sum and product

11 Disequazioni

• Compact procedure for second-degree inequalities

12 La parabola

• Two forms for finding the equation
• Vertex, axis and concavity

13 Radicali ed equazioni irrazionali

• Operations with radicals
• The least-squares formulae
• The radical and the fractional exponent

15 Similitudine

• Scale model
• Scaling laws for similar figures

16 Circonferenza

• Degree-radian conversion
• Distance of a point from a line
• Equilateral triangle
• Isosceles right triangle 45°-45°-90°
• Perpendicular bisector of a segment in coordinates
• Radians and arc length
• Radius of the circumscribed circle
• Radius of the inscribed circle
• Right triangle 30°-60°-90°

17 Disequazioni irrazionali

• Application — lines at an assigned distance
• Example — distance in a pencil of lines

18 Fasci di rette

• Complete study of a pencil
• Example — Fixed distance
• The improper pencil
• The pencil with two generators
• The proper pencil

19 Luoghi geometrici nel piano cartesiano

• Formulae for focus, vertex and directrix

20 Circonferenza nel piano cartesiano

• Equation from centre and radius
• Implicit and explicit form

21 Ellisse

• Canonical equation
• Elements of the ellipse
• Ellipse given the foci and a point
• Translated ellipses

22 Iperbole e funzione omografica

• The polynomial division trick
• Translated and non-rectangular hyperbolas

23 Funzioni, linguaggio e proprietà

• How to compute the inverse
• Price of maximum profit

24 Trasformazioni nel piano cartesiano

• Transformation from circle to ellipse
• Translated parabola

25 Funzione esponenziale

• Atmospheric pressure with altitude
• Beer-Lambert law
• Newton’s cooling
• Radioactive decay
• Radioactive decay and carbon-14 dating
• Simple, compound and continuous interest

26 Funzione logaritmica

• Logarithm using the properties
• The change-of-base formula
• The properties of logarithms

28 Circonferenza goniometrica e funzioni sin, cos, tan

• Angle of an inclined plane
• Arcs differing by pi
• Complementary arcs
• Conversion between degrees and radians
• Conversion between polar and Cartesian
• Conversion to radians
• Equations in polar coordinates
• From Cartesian to polar
• From Cartesian to polar for a circle
• From polar to Cartesian
• Fundamental identity with complementary arcs
• How to reduce an angle to the first quadrant
• Identity with arctan
• Notable values
• Opposite arcs
• Polar equation of the parabola
• Supplementary arcs
• The ladder leaning against the wall
• Values of inverse functions
• Values using associated arcs
• Worked examples of the inverses

29 Formule di addizione, duplicazione, bisezione

• 3
• Example: cosine of 15°
• Example: cosine of 75°
• Example: linearising sin⁴α
• How the other formulae are derived
• Linearising sin²α cos²α
• Lowering the degree: linearisation formulae
• Sine of 105° and cosine of 165°
• Tangent of 22.5°
• The addition and subtraction formulae
• The bisection formulae
• The duplication formulae
• Triple-angle formula for the cosine

30 Formule parametriche, prostaferesi, Werner

• Difference to product with sum-to-product formulae
• Expression with parametric formulae
• Factorising a sum into a product
• Identity on the three angles of a triangle
• Linear equations in sine and cosine
• Product to sum with Werner
• Sum-to-product formulae: from sum to product
• The parametric formulae
• Werner formulae: from product to sum
• When to use what

31 Equazioni goniometriche

• Definition and solution method
• Elementary equation in cosine
• Elementary trigonometric equations
• Equation in tangent squared
• Equation with sum-to-product formulae
• Equations reducible to homogeneous ones
• Equations solvable by substitution
• Linear equation with two methods
• Second-degree equation in sine
• Second-degree homogeneous equation
• The auxiliary-angle method
• The parametric-formulae method

32 Disequazioni goniometriche

• Factorisation with the double-angle formula
• Inequality reducible to the cosine
• The sign-study method

33 Trigonometria dei triangoli

• Angle from three sides
• Area from two sides and the included angle
• Side of an inscribed regular polygon
• Solving a triangle with the law of cosines
• Solving a triangle with the sine rule
• The chord theorem
• The law of cosines
• The sine rule
• Triangle from two angles and a side
• Trigonometric formula for the area
• Trigonometry in right-angled triangles

34 Combinatoria

• Calculating binomial coefficients
• Expansion of (2x-3) to the fifth
• Newton’s binomial theorem
• Properties of the binomial coefficient
• The four slip formulae

35 Probabilità avanzata

• Bayes’ theorem
• Complement and union of events
• Expected value
• Formula of the binomial distribution
• Product formula

36 Numeri complessi

• Euler’s identity and exponential form
• Fourth roots of -16
• Generalised Vieta’s relations
• Newton’s identities for power sums
• nth roots of a complex number
• Operations in Cartesian form
• Power with De Moivre
• Product and power in exponential form
• Quadratic equation in C
• Sum and product of the roots
• Sum of the squares of the roots
• The cube roots of 8
• The fundamental theorem of algebra
• Trigonometric and exponential form
• Trigonometric form and argument

37 Geometria analitica nello spazio

• Cartesian form of the line
• Distance between two skew lines
• Distance, midpoint and centroid
• Equation of the plane
• Equation of the sphere
• Example - distance between two points
• Improper pencil
• Improper pencil parallel to a plane
• Parallel and perpendicular component
• Parametric form of the line
• Plane through a point with given normal
• Plane through three points
• Plane through three points
• Point-line distance
• Point-line distance
• Point-plane distance
• Proper pencil
• Proper pencil through a line
• Radius of the section
• Relative positions of line and plane and angles
• Scalar product
• Sphere and plane
• Sphere centre radius and tangent plane
• Sphere-plane position
• Tangent plane to the sphere
• Unit vector
• Vector product
• Vector, magnitude and midpoint

38 Geometria sintetica nello spazio

• Cone - volume, surface area and net
• Development angle of the cone
• Frustum of a pyramid with a square base
• Planar development of the cone
• Ratios of similarity between solids
• Rectangular parallelepiped
• Regular tetrahedron
• Right pyramid with a square base
• Sphere inscribed in a cube
• Surface area and volume of the sphere
• The cylinder
• The regular tetrahedron
• Volume and surface area of the cone
• Volume and surface area of the prism
• Volume, surface area and slant height of the pyramid

39 Successioni numeriche

• AP and GP compared
• Sum of the first n terms of a GP
• Sum of the first n terms of an AP
• 3

40 Limiti di funzioni

• Determinate forms
• Equivalent functions
• Standard limits
• The trick for powers A to the B

41 Continuità e discontinuità

• Computing roots with Newton
• Cube root of 10 with Newton
• Newton iteration for the reciprocal
• Newton on x³−3x+1
• The bisection algorithm
• The Newton-Raphson method

42 Derivate

• Catalogue of composite derivatives
• Linearity, product and quotient
• Maclaurin of tan x
• Maclaurin polynomial of e to the 2x
• Standard Maclaurin expansions
• Table of fundamental derivatives
• Tangent line and normal line
• Taylor polynomial of ln x at x₀=1
• The circle diagram method
• The Taylor polynomial

44 Integrale

• Even and odd functions
• Table of elementary antiderivatives

46 Equazioni differenziali

• Solution formula for first-order linear equations
• Worked example y’ − y = eˣ
• Worked exercise — y’ − y = eˣ

47 Distribuzioni di probabilità

• Binomial distribution
• Congruences modulo n
• Cumulative distribution function
• Fermat’s little theorem and Euler’s theorem
• Mean and variance of a binomial
• Mean, variance and standard deviation
• Modular arithmetic - sum, product, power
• RSA with p=5 and q=7
• Toy RSA - p=11, q=13