Sums Learning

Introduction:

Children can be trained to count before they can learning the sums, just as they learn to talk before they can read or write. Thus, We are starting these guides to fundamental arithmetic in the context of a pre-literate person, and treating the reading and writing of numbers separately. Most seriously numerate people can do a huge deal with just the four basic operations of addition, subtraction, multiplication and division, drawing up necessary methods (formulae, or even tricks) in learning sums.

 

Counting for sums learning:

 

Numbers are a sequence of magical human grunts that are learning in order. The counting is defined as the one which is useful in our daily life. For example it starts in our date counting and ends in our stars count on the sky. The counting differs on age of the human. For the child the counting is learn through the ten fingers.

Learning multiplication, or repeated addition:

Multiplication can be regard as repeated addition, or constant addition in sums. But often, as a substitute of adding just one each time, a collection or set of items is added again and again. A convenient number of substances in a set, that has become traditional around the world, are ten. Thus, each time we assemble ten things, we tend to regard them as a set or unit. Thus the learning of multiplication is easier.

 

Writing down for sums learning

 

•First of all divide the page into two using a ruler. There is no need to have the identical measure on each side.

•In a column let we have the sums or on either column we shall have the sums if we have the ten objects or other terms.

• let you mention the sign of what sort of sums you are learning, So it may help you for later on seeing the page on which type of sums you are learning.

Summation Learning

Sigma or summation symbol:

The learning of the sigma or summation symbol from the Greek Alphabet Sigma, such as

`sum_(k=i)^n` V_i

Where,

k = is the index of the sequence of numbers.

i = starting position of index in the required sequence.

n = last position of index in the required sequence.

V_i = Variables of the i^th index positions.

Note:

The index k is always integer.

The last Index must be greater or equal to the first Index .

i.e.,      n `>=` i

 

Some important formulas from summation learning:

 

The summation notation used for the following mathematical formulas, such as

1.  `sum_(k=1)^n` k¹ = `(n(n + 1))/2`

2. `sum_(k=1)^n` k² = `(n(n+1)(2n+1))/2`

3. `sum_(k=1)^n`  k³ = `(n^2(n+1)^2)/4`

4. `sum_(k=1)^n` k⁴ = `(n(2n+1)(n+1)(3n^2+3n-1))/30`

5. `sum_(k=1)^n` k⁵ = `(n^2(2n^2 + 2n – 1)(n+1)^2)/12`

6. `sum_(k=1)^n` k⁶ = `(n(2n + 1)(n + 1)(3n^4 + 6n^3 – 3n +1))/42`

7. `sum_(k=1)^n` k⁷ = `(n^2(3n^4 + 6n^3 – n^2 – 4n + 2).(n+1)^2)/24`

8. `sum_(k=1)^n` k⁸ = `(n(2n + 1)(n + 1)(5n^6 + 15n^5 + 5n^4 – 15n^3 – n^2 + 9n -3))/90 `

9. `sum_(k=1)^n` k⁹ = `(n^2(n^2 + n – 1)(2n^4 + 4n^3 – n^2 – 3n + 3).(n+1)^2)/20`

 

10. `sum_(k=1)^n` k¹⁰ = `(n(2n + 1)(n + 1)(n^2 + n -1)(3n^6 + 9n^5 + 2n^4 – 11n^3 + 3n^2 + 10n – 5))/ 66`  etc.,

 

Examples for sumation learning:

 

Example 1:

By learning summation, fo rm the general notation of summation for the given series of addition numbers.

(a). 1 + 2 + 3 + 4 + 5.

(b). x₁ + x₂ + x₃ + x₄ + x₅.

Solution:

(a). Given:

1 + 2 + 3 + 4 + 5 = `sum_(k=1)^5` k

=`(n(n+1))/2`

=`(5(5 + 1))/2 `           (here, last index n is 5 and first index is 1)

=`(5(6))/2`

= `30/2`

= 15.     (Result)

Therefore, the generla form of the above series is `sum_(k=1)^n` k.

(b). Given:

x₁ + x₂ + x₃ + x₄ + x₅,

Here, first index is 1.

the last index is 5 and its also equal to the total elements of the given sequence, and the variable of this sequence is x.

Therefore, the general form of the above series,

`sum_(k=1)^5` x_k   = x₁ + x₂ + x₃+ x₄ + x₅.

I am planning to write more post on How to do Square Roots with example, Definition of Decimal. Keep checking my blog.

Example 2:

Form the general formula for summation of given the series of numbers by learning summation or sigma symbol.

(a) {4, 7, 5, 9, 2, 4, 7}

Solution:

Given:

{4, 7, 5, 9, 2, 4, 7}

Let x₁, x₂, x₃, x₄, x₅, x₆ and x₇ be 4, 7, 5, 9, 2, 4, 7 respectively.

That is, x₁ = 4, x₂ = 7, x₃ = 5, x₄ = 9, x₅ = 2, x₆ = 4 and x₇ = 7.

Therefore, the summation of above series are,

x₁ + x₂ + x₃ + x₄ + x₅ +x₆ + x₇ = `sum_(k=1)^7` x_k

This is required general form of summation of given sequence of numbers.

Trigonometric

Introduction :

1)  Identify the leg opposite and the leg adjacent of a specific non-right angle in a right triangle.
2) State the meaning of the each trigonometric ratio sine, cosine, and tangent, using the mnemonic device sohcahtoa if necessary.
3) Write down the ratios for the sine, cosine, and tangent for a right triangle with the lengths of the sides given.

 

Types of trignometric

 

A trigonometric equation is an equation involving  trigonometric functions of unknown angles

1) Types of Trignometry

2) Trignometric equations

3) Trig unit circle

4) Identities of trignometry

Types of Trignometry:    

we will consider two types of trigonometry:

1) trigonometry based on a circle where: a² = b² + c²

2) trigonometry based on a hyperbola where: a² = b²– c²

Trigonometry is about the angles and triangles, there are relationships between the angles and the ratios of the lengths of side of the triangles. These ratios are represented by the functions such as sine and cosine, which is occur widely in mathematics and physics, even in fields which don’t initially appear to be related to physical triangles.

I am planning to write more post on Similar Triangle Proofs with example,Special Quadrilaterals. Keep checking my blog.

    Trignometric equations :

Some of the equations which is involve trigonometric functions of the unknown may be readily solved by  the simple algebraic ideas

    Trig unit circle:

The trigonometric functions are cosine and sine may be defined on the unit circle as follows.

The Unit Circle can be defined as  a tool used in understanding sines and cosines of angles found in right triangles. It is so named because its radius is exactly of one unit

      Identities of trignometry :                                                    

The identities mostly refer to as one angle denoted t, but there are a few of them involving two angles, and for those, the other angle is denoted as s.                     

 

Properties of trignometric

 

1) The Pythagorean theorem (which is really our definition of distance as discussed below).2)

2) The addition theorems which are expressions for sin(a + b)and cos(a + b).

3) The half angle theorem (a consequence of the previous two).

4) All trigonometric functions depend only on the angle mod 2∏.

Trigonometric Formulas Learning

Introduction:

Trigonometric formulas used to express the trigonometric functions. Trigonometric individuality and formulas are based on trigonometric functions. The essential trigonometric functions are Sine, Cosine and tangent functions of a triangle takes an angle and give the sides of the triangle. Where the sin takes an angle and gives the length of the y component. Cosine role takes an angle and gives the x component length. Similarly the Tan value takes an angle and gives the slope of the triangle.

Formulas on trigonometric formulas learning

First we have to learn the basic functions which are used in trigonometric formulas

Sin rate of the angle = opposed / Hypotenuse

Cos assessment of the angle = Adjacent / Hypotenuse

Tan of the angle = Opposite / Adjacent

Formulas on trigonometric formulas learning

There are totally five types of trigonometric formulas,

1. Sum and difference formula

2. Double Angle Formula

3. Triple angle Formula

4. Half angle formulas

5. Sum of product formulas

I am planning to write more post on Special Parallelograms with example, Corresponding Parts of Congruent Triangles. Keep checking my blog.

Learning of Sum and Difference formulas:

Cos (A+B) = Cos A Cos B – Sin A Sin B

Cos (A-B) = Cos A Cos B + Sin A Sin B

Sin (A+B) = Sin A Cos B + Cos A Sin B

Sin (A-B) = Sin A Cos B- Cos A Sin B

Tan (A+B) = (Tan A + Tan B) / (1 – TanA tanB)

Tan (A-B) = (TanA- TanB) / (1+TanA TanB)

Learning of Double Angle Formula:

Sin2A = 2 Sin A Cos A

Cos 2A = Cos²A – Sin²A

Cos 2A = 1 – 2Sin² A

Cos 2A = 2Cos2A – 1

Tan 2A = 2 Tan A / (1 – Tan2 A)

Learning of Triple Angle Formulas:

Tan 3A = (3 Tan A – Tan³A) / (1 – 3 Tan² A)

Sin 3A = 3 Sin A – 4Sin³A

Cos 3A = 4Cos³A – 3 Cos A

Learning of Half Angle Formulas

Sin2 (A / 2) = (1 – Cos A) / 2

Cos2 (A / 2) = (1 + Cos A) / 2

Tan2 (A / 2) = ((1 – Cos A) / 2) / ((1 + Cos A) / 2)

Learning of Product of Sum Formulas:

Sin A Sin B = [Cos (A – B) – Cos (A+B)] / 2

Sin A Sin B = [Cos (A – B) + Cos (A+B)] / 2

Sin A Cos B = [Sin (A – B) + Sin (A+B)] / 2

Right Triangles Trigonometry Learning

Trigonometry

The word trigonometry is derived from combination of three Greek words ‘tri’ , ‘gon’ and ‘metron’.

‘Tri’ means three , ‘gon’ means sides and ‘metron’ means a measure.Thus trigonometry deals with the measurement of the sides (and angles) of triangle.

Given some sides and angles trigonometry helps in calculating remaining sides.Its Knowledge has been extensively used in astronomy, surveying, geography, navigation, physical and engineering services to determine heights and distances.

Right triangle : The triangle which contains 90 degrees as one of its angles is called right triangle. In right triangle, the side opposite to the angle 90 degrees  is called hypotenuse. Other two sides which are adjacent to 90 degrees are called legs (perpendicular legs).

Now I will go through the following trigonometry concepts using right triangles

1. Angle

2. Trigonometric ratios

3. Trigonometric Reciprocal ratios

4. Trigonometric identities

1. Angle :

Consider a ray OA. If this ray rotates about end point O and takes the position OB, then we sat that the angle <AOB has been generated. Thus, an angle is considered as the figure obtained by rotating a given ray about its end point.

In right triangle, if we take an angle between hypotenuse and one of the legs as θ, then the side which is opposite to θ is called opposite side and the side which lies in between 90 degrees and θ is called adjacent side. We know that the side opposite to 90 degrees (right angle) is hypotenuse.

 

I am planning to write more post on Find the Foci of an Ellipse with example,Perpendicular to Plane. Keep checking my blog.

 

Trigonometry :Trigonometric ratios

 

There are 6 trigonometric ratios based on ratios of sides of the right triangle. Out of these 6 ratios last three trigonometric ratios are  reciprocal ratios of first three trigonometric ratios. The first three trigonometric ratios are

     (i) Sine of angle θ is denoted by sinθ. Sinθ is Ratio of opposite side to hypotenuse

i.e., Sinθ = Opposite side: Hypotenuse

    (ii) Cosine of angle θ is denoted by cosθ. Cosθ is Ratio of adjacent side to hypotenuse

i.e., Cosθ = Adjacent side: Hypotenuse

   (iii) Tangent of angle θ denoted by tanθ. Tanθ is Ratio of opposite side to adjacent side

i.e., Tanθ = Opposite side : Adjacent side

 

Trigonometric reciprocal ratios of a right triangle:

 

These are the reciprocal ratios of above ratios, so we can call these as reciprocal ratios. Trigonometric reciprocal ratios are

   (iv) Cosecant of angle θ is denoted by cosecθ. This is the reciprocal ratio of sinθ. Cosecθ is Ratio of Hypotenuse to Opposite side

i.e., Cosecθ = Hypotenuse : Opposite side

(v) Secant of angle θ is denoted by secθ. This ratio is the reciprocal ratio of cosθ. Secθ is Ratio of Hypotenuse to Adjacent side

i.e., Secθ = Hypotenuse : Adjacent side

   (vi) Cotangent of angle θ is denoted by cotθ. This ratio is the reciprocal ratio of tanθ. Cotθ is Ratio of  adjacent side to opposite side

i.e., Cotθ =  Adjacent side : Opposite side

Resource Histograms

 A histogram is a graphical display of tabular frequencies, shown as adjacent rectangles.- (Source from wikipedia).

It  represents class intervals. And it also represents whose areas are directly proportional to the corresponding frequencies. The resource histogram is a tool which is often used by the project management team and it is a means of providing a visual representation to the team.

 

Explanation of Resource Histograms:

 

 

Explanation of Resource Histograms are given below: 

• Resource Histograms Explanation is given below:
• Resource histogram is specifically a bar chart which is used for the purposes of displays the specific amounts of the time that a particular resource which is scheduled to be worked on over a predetermined and in a specific time period.
• Resource histograms containing the comparative feature of a resource availability, which is used for comparison and for the purposes of a contrast.
•  Resource histograms are handy tools to utilize the project team otherwise the team leader, because of they allow to quick & easy way to single page view of exactly.
• What are the resources available? What are the resources is being utilized at the time now?  And how long those resources are expected to be tied up? Some of These type question’s they have to do answers.
• “It includes less than eight numbers of classes”.
• It is also called as frequency diagram.

 

Application of Resource Histogram:

 Resource histogram is used for multiple applications: some of them are given below:

• A histogram is showing statistical information that uses rectangle frequency of data items in continues numerical intervals of equal size.
• Summarizing the large data sets graphically.
• Make sure that the communication informed to the team.

 

Concepts of resource histograms with example:

 

Histograms are useful data summaries that they convey the following information:

• The common shape of the frequency distribution is  normal, chi-square, etc.
• It is Symmetrical about the distribution, where it is skewed.
• Unimodal, bimodal, or multimodal are three types of modality.

I am planning to write more post on Definition of Mode with example, Sample Variance Standard Deviation. Keep checking my blog.

Example Figure for resource Histograms:

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Learning Combined Equation

Learning Combined equation deals with an equation in which a homogeneous function is equal to zero. Combined equation relates with the Homogeneous function in which through the equation, in the given way as like the sum of the indices occurring in each term is constant. For e.g. x³ + 2x² + y³ +5xy², here degree of each term is 3. by this way learning of combined equation is proceeded.

 

Learning Combined equation – Definition

 

A combined equation of the two lines all the way through the origin is a homogenous equation of second degree. – Let the two strokes overtake through the starting point be y = m 1 x and y = m 2 x. i.e. y – m 1 x = 0 and y – m 2 x = 0 their combined equation then this is clearly a homogeneous or combined equation… ²A normal, integral, algebraic equation of the variables x and y is said to be homogeneous equation of n^th degree in x and y, when the sum of the indices of x and y in every term is the same and is equal to n.

I am planning to write more post on Inverse Functions Calculus with example, How to Solve Multi Step Inequalities. Keep checking my blog.

 

Example for learning combined equation

 

Combined equations of two outlines are in all the way through the origin as a homogeneous equation of the second degree

Let,

y = S₁x … (1)

y = S₂x … (2)

Be the lines through the origin.

The equations (1) and (2) can be written as

y – S₁ x = 0 and y – S₂x = 0

Their combined equation is

(y – S₁X)(Y- S2X) = 0

Y² – (S1 + S2)XY + S1S2X² = 0  …(3)

This is clearly a homogenous equation of the second degree in x and y.

To express ax² + 2hxy + by² = 0 as bS² + 2hS + a =0

ax² + 2hxy + by² = 0

Divide equation (1) by x²,

a+ 2h`(y)/(x)` + b(`(y)/(x)`)² = 0

Substituting `(y)/(x)` = S, we get

a+ 2hS + bS² = 0

This can be written as bS² + 2hS + a = 0

This is a quadratic form in m. This has two roots (say) S₁ and S_2.

Therefore S₁ + S₂  = `(-2h)/(b)` ,  S₁S₂ = `(a)/(b)`

 

Learning Consecutive Angles

Consider the following triangle ABC.

Initial side of /_ A is the line segment AB.

Initial side of /_ B s the line segment AB.

We observe that the above two angles have the same initial side

Such type of angles are called as consecutive angles.

Let us learn about the other consecutive angles in the same triangle.

Similarly,  /_ B and /_C are consecutive angles as they have the same initial side BC.

/_ A and /_C are consecutive angles as they have the same initial side AC.

So, consecutive angles are the two angles that have a common side as their base.

 

Learning consecutive angles of quadrilateral:

 

Quadrilateral: Quadrilateral is a closed plane figure with four sides forming four  vertices and four angles.

Let us find the  consecutive angles of quadrilateral.

According to the definition of consecutive angles, we could see that /_A and /_B share a line segment AB as one of its side.

Similarly, /_B  and /_ C  share the line segment BC as one of  it’s side

/_C and /_ D  share the line segment CD as one of  it’s side

/_A  and /_ D  share the line segment AD as one of  it’s side

The pair of consecutive angles found in quadrilateral are /_A and /_B, /_B and /_C, /_C and /_D, /_D and /_A.

There are different types of quadrilaterals, say square, rectangle, rhombus, parallelogram, trapezoid…

Square, rhombus, rectangle comes under parallelogram as all of them have their opposite sides equal and parallel.

Let us learn the properties of consecutive angles in parallelograms.

I am planning to write more post on Inverse Trigonometric Functions with example, Volume of Triangular Pyramid. Keep checking my blog.

 

Learing consecutive angles of parallelogram:

 

Parallelogram: Parallelogram is a type of quadrilateral which has equal and parallel opposite sides and opposite angles are congruent.

In the parallelogram, consecutive angles are supplementary.That is, if we add the consecutive angles, we get a sum of 180°.

Learning to solve example on consecutive angles of parallelogram:

• If one of the consecutive angle of parallelogram is 70°, then find the remaining angles of the parallelogram.

Let ABCD be the parallelogram.

Let /_A = 70°

We know that opposite angles of parallelogram are equal.

So /_C = 70°

/_A and /_B are consecutive angles.

We know that consecutive angles of the parallelogram are supplementary.

So, /_A  +  /_ B  = 180°

70° + /_ B  = 180°

/_B  = 110°

Learn Navigational Coordinates

Introduction:

Coordinates are balance used to explain a position. A lot of different kinds of learn navigational coordinate are used. Significant navigational single are depicted under. If a position is standard by a specific line, simply one coordinate is necessary to recognize the position if an origin is declared or understood. If a position is standard by a particular outside, two coordinates or balance are necessary to explain the position.

 

Navigational Coordinates

 

Let us see about learn navigational coordinates,

Every coordinate require an origin, any confirmed or implied. If a position is standard by a given plane, it power be separate by means of its reserve from every of two intersect lines, called axes. This is known as rectangular learn navigational coordinates.

In shape, OY is label the ordinate, and OX is label the abscissa. Position O is the basis, and line’s OX and OY the axis is label by the X and Y axis, equivalent. The co-ordinate position P is at location x, y. If the two axis are not perpendicular other than the lines x and y are drawing parallel to the axis, oblique navigational coordinates result. Additional kinds are label Cartesian coordinates.

Three–dimensional :

A  three–dimensional configuration of Cartesian coordinates, between X Y, and Z axes, is label space learn navigational coordinate. Other configuration of plane coordinates in common usage consists of the path and distance from the origin is labeled the limit. A geometric line extends in the ways indicate is label a radius vector. Direction and distance from a stable point stand for polar coordinates, irregularly labeled the rho–theta structure.

I am planning to write more post on Adding Binary Numbers with example, What is a Triangular Pyramid. Keep checking my blog

Spherical coordinates are use to explain a position on the outside of a area by representing pointed distance from a mainly important large circle and a position less important large circle.

 

 

 

Navigational Coordinate – Example

 

Let us see about learn navigational coordinates about the examples,

Example 1

Find the position of 2D  coordinates in diagram?

Solution :

Position of 2D coordinates is  (-2, 5).

Example 2

Find the position of 3D  coordinates in diagram?

Solution :

Position of 3D coordinates is  (5, -9, -5).

Learn Online Long Division

Introduction

Learning online long division is an essential one.Long division is the standard procedure suitable for dividing simple or complex multi digit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps.

 

Online Learning of Parts of a Long Division Equation

 

Each number has a unique name in the long division.

For example

Divide 16 by 8 gives the answer 2

Here,

The dividend is ’16’.

The divisor is ‘8’.

The quotient is ‘2’.

Here remainder is ‘0’ because there is no remainder.

Long division does not have a slash (/) and obelus (÷) operator.

 

Online Learning Example Problems for Long Division

 

Online Learning Long division without reminder:

Problem 1: Divide 548 by 4

Solution:

137      (Explanations)

4)548

4        (First number of the dividend is divided by devisor: 4 × 1 = 4)

148     (Subract the bottom number from the top number: 5 – 4 = 1)

12       (First two number is divide by divisor: 4 × 3 = 12)

28      (Subract the bottom number from the top number: 4 – 2 = 2)

28      (Previous result is divided by divisor: 4 × 7 = 28)

0       (Subract the bottom number from the top number; 28 – 28 = 0)

I am planning to write more post on Elimination Method with example, Binary Division. Keep checking my blog

Online Learning Long division with reminder:

Problem 1: Divide 749 by 4

Solution:

187      (Explanations)

4)749

4        (4 × 1 = 4)

349     (7 – 4 = 3)

32       (4 × 8 = 32)

29      (4 – 2 = 2)

28      (4 × 7 = 28)

1       (9 – 8 = 1)

Problem 3: Algebra long division. This problem will learn you how to rewrite a rational function such as

(3x³ – 2x² + 4x – 3) / (x² + 3x + 3)

Solution:

= (3x³ – 2x² + 4x – 3) / (x² + 3x + 3)

= (3x – 11) + (28x + 30) / (x² + 3x + 3)

The expression

3x – 11

is called the quotient, the expression

x² + 3x + 3

is called the divisor and the term

28x + 30

is called the remainder

 

Practice Problems for learning online Long Division:

 

Problem 1: Divide 715 by 5

     Answer: 143

Problem 2:  Rewrite the equation using polynomial long division (x³ – 1) / (x + 2)

     Answer:  (x² – 2x + 4) + (-9) / (x + 2)