High school math formulas. Mathematical formulas represent a synthesis of the development of reasoning and are made up of numbers and letters.

Knowing them is necessary to solve many problems that are charged in competitions and in Enem, mainly because it often reduces the time to solve a problem.

However, just decorating the formulas is not enough to be successful in their application. Knowing the meaning of each quantity and understanding the context in which each formula should be used is critical.

In this text we gather the main formulas used in secondary school, grouped by content.

Table of contents

The functions

Functions represent a relationship between two variables, so a value assigned to one of them will correspond to a unique value of the other.

Two variables can be associated in different ways and according to their formation rule, they receive different classifications.

Refine function

f(x) = ax + b

a: slope
b: linear coefficient

Quadratic function

f (x) = ax2+ bx + c , where ≠ 0

a, b, and c: second degree function coefficients

Roots of the quadratic function

Vertex of the parable.

Δ: discriminant of the quadratic equation ( Δ = b2 - 4.ac)

a, b and c: coefficients of the quadratic equation

Modular function

Exponential function

f(x) = ax, with a> 0 and ≠ 0

Logarithmic function

f(x) = logun x , with positive real and a 1

Sine function

f (x) = sin x

Cosine function

f(x) = cos x

Polynomial function

f(x) = an . xn + an-1. xn-1+… + A2 . x2 + a1 . x1 + a0 0

unneln-1, …, he2el1el0 0 : complex numbers
n: integer
x: complex variable

 

Progressions

Progressions are numerical sequences in which, starting with the first term, all the others are obtained by adding or multiplying by the same value.

In progressions called arithmetic, subsequent terms are found by adding the preceding term with the same number (ratio).

In geometric progressions, the sequence is formed by multiplying the previous term by the ratio.

Arithmetic progression

General term

unn = A1 + (n - 1) r

unn: General term
un1: 1st term
n: number of terms
r: BP ratio

Sum of a finite PA

Sn: sum of n terms
un1: 1st term
unn: nth term
n: number of terms

Geometric Progression

General term

unn = A1 . whatn-1

unn: nth term
un1: 1st term
q: PG ratio
n: number of terms

Sum of a finite PG

Sn: sum of n terms
un1: 1st term
q: PG ratio
n: number of terms

Limit of the sum of an infinite GP

: sum limit when the number of terms tends to infinite
un1: 1st term
q: PG ratio
n: number of terms

See also:

Plane geometry

Plane geometry is the part of mathematics that studies the properties of geometric figures in the plane. The study of geometry involves the application of postulates, axioms and theorems.

Sum of the interior angles of a polygon.

Syo = (n - 2). 180º

Syo: sum of interior angles
n: number of sides of the polygon

Story theorem

AB and CD: segments of a line determined by cutting with a bundle of parallel lines
A´B´ and C´D´: segments of another straight line, transversal to the first one, determined by cutting with the same bundle of parallel lines

Metric relationships in the right triangle

b2 = a. n

a: hypotenuse
b: side
n: projection of catheter b over the hypotenuse

c2 = a. m

a: hypotenuse
c: side
m: projection of side c on the hypotenuse

ah = b. c

a: hypotenuse
b and c: collectors
h: height relative to the hypotenuse

h2 = m. n

h: height relative to the hypotenuse
m: projection of side c on the hypotenuse
n: projection of catheter b over the hypotenuse

un2 = b2 + c2 (Pythagoras theorem)

a: hypotenuse
b and c: collectors

Polygon inscribed in the circumference.

Inscribed equilateral triangle

: measured on the side of the inscribed triangle
r: radius of the circumference

r: radius of the circumference
un3: apothem of the inscribed equilateral triangle

Registered square

: measured on the side of the inscribed square
r: radius of the circumference

un4 4: apothem of the inscribed square
r: radius of the circumference

Inscribed regular hexagon

measure on the side of the inscribed hexagon
r: radius of the circumference

un6 6: insertion of the inscribed hexagon
r: radius of the circumference

Circumference length

C = 2.π.r

C: circumference length
r: radius of the circumference

Plane figures area

Triangle area

A: area of ​​the triangle
b: measure of the base
h: height measurement relative to the base

Heron formula for the area of ​​the triangle

p: semiperimeter
a, b and c: sides of the triangle

Equilateral triangle area

A: area of ​​the equilateral triangle
measure on the side of the equilateral triangle

Rectangle area

A = bh

A: rectangular area
b: measure of the base
h: height measurement

Square area

A = L2

A: square area
L: side measurement

Parallelogram area

A = bh

A: area of ​​parallelogram
b: base
h: height

Trapezoidal area

A: trapezoidal area
B: measurement of the main base
b: measurement of the smallest base
h: height measurement

Rhombus area

A: rhombus area
D: measure of the largest diagonal
d: smallest diagonal measurement

Regular area of ​​the hexagon

A: regular hexagon area
lateral hexagon measurement

Circle area

A = π. r2

A: area of ​​the circle
r: radius measure

Circular sector area

A: area of ​​the circular sector
αrad: angle in radians
A: radius
αdegrees: angle in degrees

See more:

Trigonometry

Trigonometry is the part of mathematics that studies the relationships between the sides and angles of triangles.

It is also used in other areas of study, such as physics, geography, astronomy, engineering, among others.

Trigonometric relationships

sin: sine of angle B
b: side opposite angle B
a: hypotenuse

cos: cosine of angle B
c: side adjacent to angle B
a: hypotenuse

tg: tangent of angle B
b: side opposite angle B
c: side adjacent to angle B

sen2 α + cos2 α = 1

sin α: sine of angle α
cos α: cosine of angle α

tg α: tangent of angle α
sin α: sine of angle α
cos α: cosine of angle α

cotg α: cotangent of angle α
tg α: tangent of angle α
sin α: sine of angle α
cos α: cosine of angle α

sec α: secant of angle α
cos α: cosine of angle α

α cossec: angular cosecant α
sin α: sine of angle α

tg2 α + 1 = sec2 α

tg α: tangent of angle α
sec α: secant of angle α

cotg2 α + 1 = cosec2 α

cotg α: cotangent of angle α
α cossec: angular cosecant α

Sine Law

a: side measurement
sin: sine of the angle opposite side a
b: side measurement
sin: sine of angle opposite side b
c: side measurement
sin: sine of the angle opposite side c

Cosine law

un2 = b2 + c2 - 2.bccos

a, b and c: sides of the triangle
cos: cosine of the angle opposite side a

Trigonometric transformations

Sine of the sum of two arcs

sin (a + b) = sin a. cos b + sin b.cos a

sin (a + b): sine of addition of arc a with arc b
without a: sine of arc a
cos b: cosine of arc b
sin b: sine of arc b
cos a: cosine of arc a

Sine of the difference of two arches

sin (a - b) = sin a. cos b - sin b.cos a

sin (a - b): sine of the subtraction of arc a with arc b
without a: sine of arc a
cos b: cosine of arc b
sin b: sine of arc b
cos a: cosine of arc a

Cosine of the sum of two arcs.

cos (a + b) = cos a. cos b - sin a. sin b

cos (a + b): cosine of the sum of arc a to arc b
cos a: cosine of arc a
cos b: cosine of arc b
without a: sine of arc a
sin b: sine of arc b

Cosine of the difference of two arcs.

cos (a - b) = cos a. cos b + sin a. sin b

cos (a - b): cosine of the subtraction of arc a with arc b
cos a: cosine of arc a
cos b: cosine of arc b
without a: sine of arc a
sin b: sine of arc b

Tangent of the sum of two arcs.

tg (a + b): tangent of the sum of arc a to arc b (arcs where the tangent is defined)
tg a: tangent of arc a
tg b: tangent of arc b

Tangent of the difference of two arcs.

tg (a - b): tangent of the subtraction of arc a with arc b (arcs where the tangent is defined)
tg a: tangent of arc a
tg b: tangent of arc b

See more:

Combinatorial analysis

In combinatorial analysis we study the methods and techniques that allow solving problems related to counting.

The formulas used in this content are often used to solve probability problems.

Simple permutation

P = n!

n !: n. (n - 1) (n - 2)… 3) 2) 1

Simple fix

Simple combination

Newton's binomial

Tk + 1: General term

See also Combinatorial analysis exercises.

Probability

The study of probability allows to obtain the value of possible occurrences in a random experiment (random phenomenon). In other words, probability analyzes the "chances" of obtaining a certain result.

p (A): probability of occurrence of event A
n (A): number of favorable results
n (Ω): number of possible outcomes

Probability of joining two events.

p (AUB) = p (A) + p (B) - p (A ∩ B)

p (AUB): probability of event A or event B occurring
p (A): probability of event A
p (B): probability of event B occurring
p (A ∩ B): probability of event A and event B occurring

Probability of mutually exclusive events.

p (AUB) = p (A) + p (B)

p (AUB): probability of event A or event B occurring
p (A): probability of event A
p (B): probability of event B occurring

Conditional probability

p (A / B): probability that event A has occurred, event B
p (A ∩ B): probability of event A and event B occurring
p (B): probability of event B occurring

Probability of independent events.

p (A ∩ B) = p (A). p (B)

p (A ∩ B): probability of event A and event B occurring
p (A): probability of event A
p (B): probability of event B occurring

Statistics

In statistics, we study the collection, recording, organization and analysis of research data.

Using mathematical formulas, it is possible to know the information related to a given population from the data of a sample of that population.

Arithmetic average

MUn: arithmetic average
: sum of all sample values
n: amount of sample data

Variance

V: variance
(xyo - MUn): deviation of x values ​​from the arithmetic mean
n: amount of sample data

Standard deviation

SD: standard deviation
V: variance

See also Statistics and statistics - Exercises

Financial mathematics

Studying the equivalence of capital over time is the focus of financial mathematics, using formulas that allow us to know how the value of money varies over time.

Simple interest

J = C. i. t

J: interest
C: capital
i: interest rate
t: application time

M = C + J

M: quantity
C: capital
J: interest

Compound swears

M = C(1 + i)t

M. quantity
C: capital
i: interest rate
t: application time

J = M - C

J: interest
M: quantity
C: capital

spatial geometry

Spatial geometry

Spatial geometry corresponds to the area of ​​mathematics that is responsible for studying figures in space, that is, those that have more than two dimensions.

Euler relation

V - A + F = 2

V: number of vertices
A: number of edges
F: number of faces

Prisma

d: diagonal of the paver
a, b and c: measurements of the dimensions of the paver

V = B.h

V: prism volume
B: base area
h: height of the prism

Pyramid

V: volume of the pyramid
B: base area
h: height of the pyramid

Pyramidal trunk

V: volume of the pyramidal trunk
h: height of the pyramidal trunk
B: area of ​​the largest base
b: area of ​​the smallest base

Cylinder

UnL= 2.π.Rh

UnL: lateral area
A: radius
h: cylinder height

UnB = 2.π.R2

UnB: base area
A: radius

UnT = 2.π.R (h + R)

UnT: total area
A: radius
h: height

V = π.R2.h

V: volume
A: radius

Cone

UnL = π.R. g

UnL: lateral area
A: radius
g: generatrix

UnB = π.R2

UnB: base area
A: radius

UnT = π.R. (g + R)

UnT : total area
A: radius
g: generatrix

V: volume
UnB: base area
h: height

Cone trunk

UnL = π.g (R + r)

UnL: lateral area
g: generatrix
R: major radius
r: smaller radius

V: volume
h: height
R: major radius
r: smaller radius

Sphere

A = 4.π.R2

A: area of ​​sphere
A: radius

V: volume of sphere
A: radius

See more:

Analytic geometry

In analytical geometry we represent lines, circles, ellipses, among others in the Cartesian plane. Therefore, it is possible to describe these geometric shapes using equations.

d (A, B): distance between points A and B
x1: abscissa of point A
x2: abscissa of point B
y1: abscissa of point A
y2: abscissa of point B

m: slope of the line
x1: abscissa of point A
x2: abscissa of point B
y1: abscissa of point A
y2: abscissa of point B

General equation for a straight line.

ax + by + c = 0

a, b and c: constants

Reduced linear equation

y = m x + b

m: slope
b: linear coefficient

Line segmentation equation

a: value at which the line intersects the x-axis
b: value at which the line intersects the y-axis

Distance between a point and a line

d: distance between point and line
a, b and c: coefficients of the line
x: abscissa point
y: ordinate of the point

Angle between two lines

m1: slope of line 1
m2: slope of line 2

Circumference

Circumference equation

(x - xc)2 + (and - andc)2 = R2

x and y: coordinates of any point that belongs to a circle
xc yyc: coordinates of the center of the circle
A: radius

Normal equation of circumference

x2 + Y2 - 2.xc.x - 2.yc.y + (xc2 + Yc2 - R2) = 0

x and y: coordinates of any point that belongs to a circle
xc yyc: coordinates of the center of the circle
A: radius

Ellipse

(the major axis belongs to the x axis)

x and y: coordinates of any point that belongs to an ellipse
a: measurement of the semi-principal axis
b: measurement of the minor half axis

(the major axis belongs to the y axis)

x and y: coordinates of any point that belongs to an ellipse
a: measurement of the semi-principal axis
b: measurement of the minor half axis

Hyperbole

(the real axis belongs to the x axis)

x and y: coordinates of any point that belongs to a hyperbola
a: measure of the real semi-axis
b: measure of the imaginary semi-axis

(the real axis belongs to the y axis)

x and y: coordinates of any point that belongs to a hyperbola
a: measure of the real semi-axis
b: measure of the imaginary semi-axis

Parable

y2 = 2.px (vertex at the origin and focus on the abscissa axis)

x and y: coordinates of any point that belongs to the parabola
p: parameter

x2 = 2.py (vertex at the origin and focus on the ordinate axis)

x and y: coordinates of any point that belongs to the parabola
p: parameter

Complex numbers

Complex numbers are numbers made up of a real and imaginary part. The imaginary part is represented by the letter ie indicates the result of the equation i2 = -1.

Algebraic form

z = a + bi

z: complex number
a: real part
bi: imaginary part (where i = √ - 1)

Trigonometric form

z: complex number
ρ: complex number module ()
Θ: argument z

(Moivre formula)

z: complex number
ρ: complex number module
n: exponent
Θ: argument z