Vertices

Vertices:

Introduction on what are Vertices:
Before we get into the details of vertices let us first understand the meaning of a vertex.
Vertex typically means a corner or a point where lines meet. For example a square has four corners, each is called a vertex. The plural form of vertex is vertices. (Pronounced: "ver - tiss- ease"). A square for example has four vertices.

The vertices are the common factor to the point what are the two line segments are assembled. It is also distinct as the point where two sides meet. It is like corners in 2D and 3D shapes. Vertices are the plural structure of vertex.

It is obtainable only in shapes. It is not obtainable in circle and oval. The vertex forms of an angle since it is the gathering point of two lines or sides. And the angle is called as center angle.

What are Vertices some Types:

These are the some example what and how much vertices that its having,

Example 1:Polygons.

Example 2:Polyhedrons.

Example 3:Parabola.

Example 4: Hyperbola.

Example 5:Ellipse

Explanation for What are Vertices:


Polygons:

Vertex of a polygon is the points what the two neighboring sides meet are. The number of sides is equivalent to their each vertex. It is also named as Edge. It forms an angle called inner angle. Polygons are named as 2D shapes.

Example:

The number of vertices in rectangle is 4 and the number of vertices in hexagon is 6.

Polyhedrons:

In polyhedron vertex is the point what are the two or more edges meet. Based on the vertex the polyhedron is separated as regular and irregular. If two equal sides meet then it is called as Regular polyhedron. Polyhedrons are the 3D shapes.

Example:

Cube has 8 vertices and sphere has no vertex.

There are commonly three shapes in it .They are Parabola, Hyperbola and Ellipse. In this conic sections the vertex contains the points are (x, y).

Parabola:

The parabola vertex is the point what is made and its sharpest roll. It is either uphill or down. The parabola also having only one vertex.

Hyperbola:

In hyperbola vertex lies on the major axis and it occurs at the point what has its sharpest turn. The hyperbola contains only one vertex.

Ellipse:

It is identical as that of hyperbola. It dishonesty on the main axis. The ellipse also contains one and only vertex. It is also defined as in axis. They are represented as horizontally and vertically.


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Tangent Cotangent Cosecant

Tangent Cotangent Cosecant:
Let us learn about the meaning of Tangent,Cotangent and Cosecant in brief.
Definition of Tangent Cotangent Cosecant:

Some of the Trigonometric functions are defined from the right-angled triangle.

Tangent (Tan):

The ratio of length of the adjacent side and the opposite side of an angle is called as tangent.

Tan (θ) = adjacent / opposite

Cotangent (cot):

The ratio of length of the opposite side and the adjacent side of an angle is called as cotangent.

Cot (θ) = Opposite / Adjacent

Cosecant (csc):

It is the ratio of length of the hypotenuse and the adjacent Side of an angle is called as cosecant.

Cosec (θ) = hypotenuse / adjacent.

Example Problems for Tangent Cotangent Cosecant:

Example 1:

Find the measure of the length of other sides and also find the tangent function values for the given right angle triangle.


Using the trigonometry functions, find the length of the other side

Tangent function:

tan θ = adj / opp

tan θ = 8 /6

tan θ = 4/ 3

θ = tan -1 (4/3)

θ = 53 °.

Using the Pythagorean Theorem

In the given right angle triangle

AC2 = AB2 + BC2

Here,

AB = Opposite side

BC = Adjacent side

AC = Hypotenuse

AC2 = AB2 + BC2

= 62 + 82

= 36 + 64

AC2 = 100

AC = 10

Hypotenuse for the given right angle triangle is 10.

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Numbers in Scientific Notation

Numbers in Scientific Notation:

Let us learn how to write the numbers in scientific notation form.

The speed of light is 300,000,000 m/sec.

This can be written in scientific notation as 3 x 10 to the power of 8 or as 3 x 10^8.

The numb er 134000000000 can be written in scientific notation as 1.34 x 10^11.

The first number 1.34 is called the coefficient. It must be greater or equal to1 and less then 10.The second number is called he base. It must always be10and is always be written in exponent form in scientific notation.

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Simple Interest

Simple Interest:

We would have all come across the term "Interest" Interest in a lay mans language would mean to develop a liking,example:Ones Interest in collecting paintings,but in mathematics Simple interest = (pnr)/100.Let me explain this Concept Of Simple Interest more accurately:


Simple interest = (pnr)/100

Compound interest = p (1 + (r/100) n

Where,

p – Principal

n – Number of years

r – Rate of interest per annum.

The amount borrowed is called Principal. What we pay back to the money lender for using his money for a definite period is called interest or simple interest. The interest paid keeping Rs 100 for one year is called rate percent of interest. The sum of Principal and interest is known as Amount. Simple interest is generally written as interest only. So also unless it is clearly mentioned otherwise the rate of interest means Annual rate.As we have seen in the previous Blogs we get a better understanding of a topic when we solve few example problems,let us now look at few example problems on Simple Interest.



If the interest is calculated for the entire period separately for the principal the interest is called simple interest.

1. Find the simple Interest on a sum of Rs. 800 for 2.5 years at 4.5% per annum.

Solution:

Principal (p) = Rs. 800

Time (n) = 2.5 years

Rate (r %) = 4.5% per annum

Simple interest = (p x n x r)/100


= (800 x 2 .5 x 4 .5) /100

= Rs 90

Simple interest = Rs 90

2. John invested Rs. 5.000 for 3 years at the rate of 6 % per annum. Find the simple interest and the amount received by him at the end of 3 years.

Solution:

Here Principal. p = Rs.5000

Number of years, n = 3

Rate of interest, r = 6 %

Simple interest, I = (p x n x r)/100

=5000x 3 x (6/100)

= Rs. 900

Amount, A = Principal + Interest

= $5000 + 900

= Rs. 5900

The simple interest = Rs .900

Amount = Rs.5, 900


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Conic Sections:

Conic Sections:

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called the focus, and some line, called the directrix.Algebra is the one of the most important chapter in mathematics subject. Here the Conic section is one of the topics in algebra chapter. In mathematics a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. The conic sections were named and studied as long ago as 200 BC.
Let us now learn the definition of a Conic Section:
Conic section is generally defined as intersection of a plane and a cone, it will depend on How the plane is oriented, the curve will be one conic sections. Or a conic section is a curve formed by the intersection of a cone with a plane.

Four types of conic sections are there,

* Circle,
* Parabola,
* Ellipse, and
* Hyperbola.

Explanation:

* A circle is the set of all points; having the area is equal distance from a fixed point C, the midpoint (center) of the circle.
* An ellipse is the set of all points; and it is surrounded two foci, (or) focus points, such that the sum of the distances from any point to each focus remains constant. An ellipse can be oriented vertically or horizontally.
* A parabola is the set of points; and that are equal distance from the focus point and the directrix, a fixed line. A parabola can be oriented vertically or horizontally.
* A hyperbola is the set of all points; it’s around two foci, or focus points, such that the difference of the distance from any point to each focus all remains constant. A hyperbola can be oriented vertically or horizontally

General Equation for a Conic Section:

General equation for conic section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
The conic sections could involve these all:

1. The description of cones apart from the coordinates
2. The description of cones with coordinates, so that this is through equation or formulas for them.
3. The choice from the orthogonal coordinate system is (x, y, z) such that the plane, so it is corresponding to Z = 0, and a description of the cone with in coordinate system.
4. The transformation of quadratic equation is standard form by a combination of rotation.
5. Using the equation in standard form to classify the intersection as an ellipse, parabola, hyperbola etc...
6. And It has the equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

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Sequence

Sequence:

Let us understand the meaning of Sequence in general,in mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.Let us now look at a more diversified explanation about a Sequence,A set of numbers arranged in a definite order according to some definite rule is called a sequence.
or

A sequence is a function whose domain is the set N of natural numbers.
It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n Î N under 'a' by an or tn.
Examples:

1, 3, 5, 7…..... (adding 2 to every term)

1, 4, 16, 64 … (Multiplying by 4 every term)

20, 17, 14 … . (add -3 to every term)

The different numbers in a sequence are called terms of sequence.

The subscripts denote the position of the term.

In the second example, 4 is the second term, and 14 is the third term in the third example.

The nth term of a sequence is called the general term of the sequence and is usually denoted by an or tn.

Finite and Infinite Sequences:

A sequence is called finite if the number of terms is finite. A finite sequence has always a last term.
Examples:

2, 5, 8, 11, 14 …, 32
37, 33 …, 1

A sequence is called infinite if the number of terms is infinite. An infinite sequence has no last term. In this sequence, every term is followed by a new term.
Examples:

i) A sequence of multiples of 5
5, 10, 15, 20, …

ii) A sequence of reciprocals of positive integers

The above two sequences are clearly the infinite sequences.
Series

Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.

Examples:

i) 1 + 4 + 7 + 10 + ... is a series in which first term is 1, second term is 4, third term is 7 and so on.
ii) 3 - 9 + 27 - 81 + ... is also a series in which the first term is 3, second term is -9, third term is 27 and so on.


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Uniform Random Distribution

Uniform Random Distribution

Introduction to Uniform Random Distribution:

Uniform random distribution is the one of the important topic in the probability distribution chapter in mathematics subject. The simplest distribution is the uniform random distribution; The uniform random distribution is suitable for most sensitivity testing and is selected by default.Now let us learn about the Types of Uniform Random Distribution, it is classified into two types; one is discreet uniform distribution and another one is continual uniform distribution.

Uniform Random Distribution:


In order to analyses numerical data, it is necessary to arrange them systematically. An arrangement of data in a systematic order is called a uniform distribution. A uniform distribution, sometimes called as a rectangular distribution, in this distribution that has the constant Probabilities occurred.

Continuous & Discrete Uniform Distributions:

Continual uniform distributions:

It is a statistical distribution for which the variables take on continual range.There are certain phenomenon which by the lack of precision in measurement are not capable of exact measurement.
Ex: weight, height, temperature, age, etc.,
Such a series are called as continual distributions.

Discrete uniform distributions:

It is also the statistical distribution where the variables can take on only discrete values. A discrete distribution is formed from items which are capable of exact measurement.
A discrete distribution with probability function p(xk) defined over, k = 1,2...N., Has distribution function.

D (xn) = [sum_(k=1)^n] p(xk)

and population mean, is [mu] = 1 / N [sum_(k=1)^n] xk P(xk)

Ex: we can count the number of Parsons salaries are exactly Rs 100 p.m, Rs 105 p.m., or Rs 110 p.m. Other examples of discrete variables are the number of children in a family, goals scored in foot ball matches.
Uniform Random Distribution General Formula:

The general formula of probability density function of the uniform ramdom distribution function is defined as follows:

f(x) = 1 / B-A for A [<=] x [<=] B

Where A is the location parameter and (B - A) is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform random distribution.

The equation of the standard uniform random distribution is

f(x) = 1 for 0 [<=] x [<=] 1.

These all are important in the Uniform random distributions.


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