GEOMETRIA ANALÍTICA

Let us know a bit of analytic geometry.


ANALYTICAL GEOMETRY



 Hyperbolas are one of the fronts of the study Analítica.A Analytical Geometry geometry, also called geometric coordinates, based on studies of the geometry through the use of algebra. Initial studies are attached to the French mathematician René Descartes (1596 -1650), creator of the Cartesian coordinate system.

The studies related to Analytical Geometry date its beginnings in the seventeenth century, Descartes, to relate algebra to geometry, mathematical principles that created by geometric methods to analyze the properties of the point, the line and the circle, determining distances between them, location and coordinate points.

 An important feature of G.A. presents the definition of geometric shapes numeric mode, extracting information data representation. Based on these studies, mathematics is seen as a modern discipline, able to explain and demonstrate situations related to space. The intuitive notions of vectors begin to be explored forcefully in the search for numerical results that express the ideas of the union of geometry with algebra.


The vectors are the basis of studies of the vector space, objects that have the characteristics related to size and direction. The vectors are widely used in physics, as an aid in the related Vector Kinematics, Dynamics, Electric Field and other related content calculation tool.
Scientists Isaac Newton and Gottfried Wilhelm Leibniz studies focused on analytic geometry, which served as theoretical and practical for the emergence of the Differential and Integral Calculus based, very used in Engineering.

We can relate the following topics to the study of GA:

Analytical Study Point
Cartesian plane
Distance between two points
Midpoint of a segment
Alignment condition of three points
Study of Straight
And reduced overall equation of the line
Intersection of straight
parallelism
perpendicularity
Angles between straight
Distance between point and line
Study Circle
And reduced overall equation of the circle
Relative positions between dot and circle
Relative positions between straight and circumference
Problems related to tangency
Study of Conical
ellipse
hyperbole
Parable intersection between conical
Tangent lines to a conic

Exercícios de equação de segundo grau

Good afternoon friends, since my companion blog posted quadratic equations there are some exercises to fix.

Exercises Equations of 2nd Grade
1) Identify the coefficients of each equation and tell whether it is complete or not:
a) 5x2 - x3 - 2 = 0
b) 3x2 + 55 = 0
c) x2 - 6x = 0
d) x2 - 10x + 25 = 0

2) Finding the roots of equations:
a) x2 - x - 20 = 0
b) x2 - 3x -4 = 0
c) x2 - 8x + 7 = 0

3) among the numbers -2, 0, 1, 4, which ones are roots of the equation x2-2x-8 = 0?

. 4) The number -3 is a root of the equation x2 - 7x - 2c = 0 these conditions, determine the coefficient c:


5) If you multiply a real number x by itself and subtract 14 from the result, you will get five times the number x. What is this number?

Equações trigonometricas

Trigonometric equations are equalities that involve trigonometric functions of unknown arcs. The resolution of these equations consists of a single process that uses techniques of reduction to simpler equations. We will address the concepts and definitions of the equations in the form = cosx.

Trigonometric equations in the form α = cosx has solutions in the interval -1 ≤ x ≤ 1 The determination of the values of x that satisfy this equation type obey the following property:. If two arcs have equal cosines, then they are côngruos or replementares.

Consider x = α a solution of the equation cos x = α. Other possible solutions are to côngruos α bow or archery bows - α (or arc 2π - α). So: cos x = cos α. Note the trigonometric representation in the cycle:

We conclude that: 

x = α + 2kπ with k Є Z or x = - α + 2kπ with k Є Z 

example 1 

Solve the equation cos x = √ 2/2. 

By the trigonometric ratios table we have √ 2/2 corresponds to an angle of 45 º. then: 

cos x = √ 2/2 → cos x = π / 4 (π / 4 = 180/4 = 45) 

Thus, the equation cosx = √ 2/2 has as a solution to all côngruos arc π / 4 or-π / 4 or arcs 2π - π / 4 = 7π / 4. Note illustration:
We conclude that the possible solutions of the equation cos x = √ 2/2 are:
x = π / 4 + 2kπ with k Є Z or x = - π / 4 + 2kπ with k Є Z

Equação de segundo grau

Fórmula de Bhaskara

An equation is a mathematical expression that has in its composition unknowns, coefficients, exponents, and an equal sign. The equations are characterized according to one of highest power unknowns. see: 

2x + 1 = 0, the exponent of the unknown x is equal to 1. Accordingly, this equation is ranked as the 1st grade. 

2x ² + 2x + 6 = 0, we have two unknowns x in this equation, one of which has the largest exponent, determined by 2. This equation is classified as 2nd degree. 

x ³ - x ² + 2x - 4 = 0, in which case there are three unknowns x, where the largest exponent equal to 3 determines that the equation is classified as the third degree. 

Each model equation has a form of resolution. Work as a resolution of an equation of the 2nd degree, using the method of Bhaskara. Determine the solution of an equation figure is the same as the roots, that is, the value or values ​​that satisfy the equation. For example, the roots of the 2nd degree equation x ² - 10x + 24 = 0 are x = 4 or x = 6 because: 

Substituting x = 4 in the equation, we have: 

x ² - 10x + 24 = 0 
4 ² - 10 * 4 + 24 = 0 
16-40 + 24 = 0 
-24 + 24 = 0 
0 = 0 (true) 

Substituting x = 6 into the equation, we have: 

x ² - 10x + 24 = 0 
6 ² - 10 * 6 + 24 = 0 
36 - 60 + 24 = 0 
- 24 + 24 = 0 
0 = 0 (true) 

We can see that the two values ​​satisfy the equation. But how do we determine the values ​​that make the equation a true sentence? It is this way of determining the unknown values ​​that we discuss below. 

We will determine the method of resolving the values ​​of Bhaskara equation of the 2nd degree: x ² - 2x - 3 = 0. 

Tabela do litro

Table of the liter

important relationships

1DM ³ = 1l = 1kg

1m ³ = 1t = 1kl

1cm ³ = 1ml = 1g

Teorema de Pitágoras

      The Pythagorean theorem is considered one of the major discoveries of mathematics, it describes a relationship in the triangle. Remember that the triangle can be identified by the existence of a right angle, ie, measuring 90 °. The triangle consists of two peccaries and the hypotenuse, which is the largest segment of the triangle and is located opposite the right angle. Note: 

Cathetus: a and b 
Hypotenuse c 

The theorem says that: "the sum of the squares of the legs equals the square of the hypotenuse." 

a² + b² = c²

example 1 
Calculate the value of the unknown segment in the triangle below. 

x² = 9² + 12²

x² = 81 + 144

x² = 225

√x² = √225

x = 15

  It was through the Pythagorean theorem that the concepts and definitions of irrational numbers began to be introduced in Mathematics. The first to emerge was irrational √ 2, which appeared the hypotenuse of a right triangle with the other two sides measuring 1 See be calculated.: 

            

x² = 1² + 1²

x² = 1 + 1

x² = 2

√x² = √2

x = √2

√2 = 1,414213562373....

example 2
Calculate the value of the collared peccary in the rectangle triangle below:

x² + 20² = 25²

x² + 400 = 625

x² = 625 – 400

x² = 225

√x² = √225

x = 15

Suppose that 1 or Cateto Cateto A, was equal to 8 cm and the peccary or Cateto 2 B, is equal to 6 cm.Qual is the value of the hypotenuse? 

        

Solution: 

x² = 8²+6²

x² = 36+64

x = √100 = 10

x= 10 cm 

   Observation:


He knows the legs, but he does not know the hipotenusa.Então this calculation to find out the result of x ², which in this case is 10 cm.

    

Metro cúbico

The cubic meter (symbol: m³) is a unit of measurement equivalent to one thousand liters in volume. It is the standard in the International System of Units and is derived from the meters, equivalent to the volume of a cube with edges one meter. The cubic meter is equivalent to a kiloliter.















The cubic meter per square meter is different, because the cubic meter occupies three houses in the cubic square table and only two houses in each unit.

Volumes

Volume of a sphere: 4/3.3, 14, r ³

Volume of a cube: aresta.aresta.aresta

Volume of the parallelepiped: comprimento.largura.altura

Trigonometria

The Trigonometry (triangle: triangle and flowmetry: measurements) is the study of mathematics responsible for the relationship between the sides and angles of a triangle. In right triangles (have an angle of 90 °), the relations are called remarkable angles, 30 º, 45 º and 60 º, which have constant values ​​represented by sine relations, cosine and tangent. Triangles have no right angles, conditions are adapted in finding the relationship between the angles and sides.
       
         Example: 

Table of Trigonometric reasons

      

Álgebra

Algebra is the branch of mathematics that studies the equations, mathematical operations, polynomials and structures algébricas.Hoje will see the equations of first degree ie in which only contains one unknown
To solve we use the inverse operation, then we should apply the amount that the unknown side (x, y, z) to be the number of the denominator and the numerator the result of the inverse of the fraction
and then divide and find the value of x
example:

1-3x-2 = 16
    3x = 16 +2
    3x = 18
    x = 3/18
    x = 6



2 - 12-3x 30 =
      3x = 30 +12
      3x = 42
      x = 42/3
      x = 14


3 - 3x +12 = 30
     3x = 30-12
     3x = 18
     x = 3/18
     x = 6


              We may also use other results not only be 3x, we can use 2x, 4x, 5x, etc ...