First, Second and Third Person in English

First person, second person and third person is one of the most important concepts in English learning. Grammatical person in English linguistics refers to a set of personal pronouns such as I, we, you, he, and she and so on. Let’s have a closer look at each of the grammatical persons – first, second and third, along with relevant examples in this post.

First Person
‘I’ is first person singular and ‘we’ is first person plural. First person refers to a narrator who speaks being a direct character. For example: I have tried shopping kids’ shoes online and I must tell you it was a nice experience and also the kids’ shoes online collection was huge. (Here, the speaker is speaking in first person.

Second Person
When a person is speaking as a second person, he or she is directly addressing the audience. The only three examples of second persons in English language are you, your and yours. For example: Did you know that online baby stores bring really good stuffs. You get kids’ dress from popular brands like Little Kangaroos India collection and kids’ toys from Little Tikes India collection. Also you will not believe the amount of discounts available for Little Kangaroos India brand and Little Tikes brand. (Here, the speaker is addressing the audience directly and is in second person.

Third Person
Third person as the term suggests is that where a speaker talks about some third person. Examples of third persons are he, she, it, him, her, it and more. Singular third person is he, she and it and plural third person are they, the, these, that and so on. For example: Mary is a school teacher. She got married in an early age and has a baby at 26 now. She buys almost every essential for her baby through online shopping. Online shopping is the latest trend of market and online shopping for kids has entered in the life of parents as savior. (Here, the speaker is talking about some third person to the audience.)
These are the basic learning on narrator, narration and narrative.

Conjunctions and its Types

Conjunction is one of the eight parts of speech in English grammar, the other seven being noun, pronoun, verb, adverb, adjective, preposition and interjection. Let’s have a look at conjunction and its types along with examples used in combining sentences, words and phrases in this post. Conjunction is a part of speech that combines two words, phrases or sentences together. For example:
Maria bought flora pencils for her daughter
Maria bough curved scissors for her daughter.
Maria bought flora pencils  and curved scissors for her daughter.  (Here, ‘and’ is the conjunction that connects the two sentences together.
There are two broad classifications of conjunction – Coordinating Conjunctions and Subordinating Conjunctions.

Coordinating Conjunctions: 
When a conjunction is used to combine two meaningful sentences of equal importance, it is called coordinating conjunctions. Coordinating conjunctions are also termed as coordinators. In English grammar, the coordinators to combine sentences are for, and, nor, either, neither, but, or, yet, so etc. For example:
• She will either buy Kittens shoes: or Disney shoes for her baby.
• Neither he drinks nor does he smoke.
• He is poor but humble.
• Online shopping is easy yet affordable.
• She brought apples and oranges from the market.

Subordinating Conjunctions:
When a conjunction combines two sentences where one of the sentences depends on the other to convey a complete meaning, it is called subordinating conjunctions. Subordinating conjunctions are also called subordinators. Subordinators combine and independent clause and a dependent clause to convey a complete meaning. Some of the most popular subordinators used in English grammar are: after, although, as, as far as, as long as, as soon as, because, since, whenever, whereas, while, since and more.  For example:
• As soon as she heard of online baby shopping, she rushed to buy baby food.
• Whenever he comes, it rains.
• Since I left the city, I have never been there again.
• While I was shopping online, I availed a discount coupon of branded stuff.
• He has not come today because he is ill.
These are some of the basics on conjunction and its types.

Understanding the laws of Boolean Logic

There have been various laws in the arithmetic number system. The study of these laws aids in the solving of various mathematical problems. The Boolean algebra laws are the laws concerning the logic 0 and 1. Only zero and one are used in the Boolean algebra. Sometimes true or false is also used to represent the same logic. Ultimately they represent nothing but on and off of supply of flow of current in an electric circuit.

The operations used in this algebra are similar to the laws in the algebra. There are certain basic differences and the notations might change. But the process is similar and simple. The Boolean laws can be conjunction, disjunction and negation. Conjunction is nothing but the process of multiplication that is present in the real numbers algebra. Dis junction is the process similar to addition that is done in the real number algebra. Negation is nothing but taking the negative of the given value. If the given value is x, then it is replaced by ‘-x’ in the process of negation.

This algebra was developed by Boole in the year 1840 and that is why it is known as Boolean algebra. It is used in digital electronics. It is very helpful in the analysis of various gates and circuits. It has certain laws and these have to be learnt to understand and solve the problems. The Boolean laws are to be learnt for this and they are quite simple to learn. There can be different variables used in this algebra but these variables can have only two values zero and one. A large expression can be formed with the help of the variables.

These are to be solved with the help of laws of Boolean algebra and understood. The variables like A, B, X, Y can be used to represent the variables in this algebra and then the expressions formed and the logical operations carried out. Any variable added to ‘1’ gives one. This is one of the laws. Any variable added to zero gives the same variable. Any variable multiplied with ‘1’ gives the same variable. The same variable if multiplied with ‘0’ gives ‘0’. In electronics it represents an open circuit. Two similar variables on addition give the same variable. Multiplication of two similar variables gives the same variable. Commutative property of addition also holds good. These are some of the laws. The variables can take values 0 and 1 only.

Department of education

The department of education in math gives us various contents and syllabus. The math can be differentiated into various topics along the grade of the class. The department of education shows the step by step development in grade wise for the math. Algebra has differentiation when compared to the grade 5 and grade 12. The math can be classified into various topics as the education depends.

Department of Education:

Let us have the topic named partial product which is made to differentiated based on the grade. The lowest grade product gives the multiplication table they are,

1 x 2 = 2.

9 x 4 = 36.

As they have shown for the higher grade the multiplication can be made as the partial product math. These give the suitable example for the department of education of math.

The partial products math is nothing but the summing of the two terms not only through the addition but also through the product method. First the terms are rounded and made to multiply with the left most term of the number. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term. Then the right most term can be made to multiplied with the right most term of the another term.

At last the values are made to summing up and the total of the values gives the product of the both terms. This is how the product partial terms are executed. The partial math includes all the operations like addition and subtraction in the same manner. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term.

Example Problems for Department of Education:

Example 1: Find the partial product of the term 93 x 25?

93

25

--------

90*20 -> 1800

90*  5 ->   450


3*20 ->     60

3*  5 ->     15

---------

2325

Answer: 2325

Writing and simplifying algebraic expressions

What is an algebraic expression?
A symbol in algebra that is supposed to have a fixed value is called a constant, where as any other symbol in algebra that can be assigned different values are called a variable. For example 1/3, -8, pi, e etc are all constants and x, y, z, etc are all variables.
A sensible combination of constants and variables conjoined by arithmetic signs of +, -, * and / is called an algebraic expression. The parts or terms of an algebraic expn are separated by + or – sign. The constants and variables that are connected by * or / signs are deemed as one term. For example, in the algebraic expression 2x + 3xy + 5y - 7 there are four terms. 2x, 3xy, 5y and -7.

Simplifying an algebraic expression:
Simplification of an algebraic expression involved addition, subtraction, multiplication and division. So, How to Simplify an Expression? The rule of PEMDAS that we follow for simplifying arithmetic calculations is also followed in simplifying algebraic expn as well. Like terms can be added or subtracted together. Like terms are the terms that have the same set of variables with same exponents.  For example, 2x, 5x, 0.75x etc are all like terms. They can be added or subtracted and combined to one single term. Whereas, 2y, 5x, 0.75z^2, x^2 etc are all unlike terms. They cannot be added or subtracted. If an algebraic expn has brackets, then they have to be simplified first. There is no clear cut step wise process to simplify algebraic expn as there can be innumerable different types of expressions, and each can be simplified in simple ways. Let us look at some examples to understand better.

Example 1: 
Simplify: 3x + 4y – 3 + 4x + 7y + 8
Solution: 
For this type of expression it is possible to simplify by combining like terms.
Step 1: collect the like terms together
=> 3x + 4x + 4y + 7y + 8 – 3
Step 2: now combine the like terms
=> (3x+4x) + (4y +7y) + (8-3) = 7x + 11y + 5
That is your answer.

Writing an algebraic expression:
For any given situation described in words, that involves numbers, we can write the corresponding algebraic expn.
Example: 
Write an expression for “twice a number added to 1”.
Solution:
Here, suppose the number is x. Then twice the number would be 2x and that when added to one gives us 2x+1. This would be our required algebraic expn.

Parabola Equation

A parabola is a conical section formed by the intersection of a conical right circular surface and plane which is parallel to the straight line generated at that surface. The parabola can also be generated by examining the point called as focus and the line called as directrix. The locus of all points in the plane which are at equal distances from both the point and the line is said to be as the parabola. The line which passes through the focus (line which splits parabola in the middle) and lies perpendicular to the directrix is known as the “Axis of Symmetry”. Also the point present on this symmetry of axis which will intersect the parabola is known as the “Vertex”. In this vertex point, the curvature will be always greatest. A parabola can be opened up and down, left and right or even in some other arbitrary directions. They can be rescaled or repositioned in order to fit exactly with any other parabola, which implies that all parabolas are similar.

Standard form of a Parabola Equation
The general form for finding the Equation of a parabola is given as,
Y = ax2 + bx + c, where ‘x’ and ‘y’ are the points on the parabola.
In the above equation, when the value of ‘a’ is greater than zero, then the parabola will open upwards and when the value of ‘a’ is lesser than zero, then the parabola will open downwards. Also, the axis of symmetry will be the line of ‘x’ value equaling to negative of ‘b’ divided by 2a.

Parabola Equation Vertex
The point where the parabola will cross its axis is simply said as the vertex of a parabola equation. From the above standard equation of parabola, it implies that when the coefficient of x2 term occurs as positive, then the vertex will be at lowest point drawn on the graph. Similarly, when comes with negative coefficient of x2, it will be at highest point which can be said to be at the “U’ shape top.
The vertex form of a parabola equation can be written as,
Y = a(x-h) 2 + k, where ‘h’ and ‘k’ are the vertices of a parabola.

The parabola can also form into horizontal direction in the graph extending through the horizontal axis. The horizontal parabola Equation is simply as same as the standard form of parabola equation.

There are numerous websites which provide Parabola Equation solver in which when we give the vertex and focus of the parabola, it will automatically generate the standard and vertex form of the parabola.

Concept of dependent variables in math and statistics

A dependent variable in math is a variable the value of which depends on one or more other variables. For example if we have an equation that looks like: y = 2x+ 3. Here, y is one such variable because the value of y would depend on what value is assigned to x. Such an equation is called an equation in two variables. When plotting such a relationship on a graph, the independent of the variable x is usually plotted on the x axis and the dependent-variable axis is usually the y axis. Therefore, if the relationship is like this: p = 3q + 7, then the independent of the variable q would be plotted on the x axis and the dependent of these which is p would be plotted on the y axis.

Dependent variable in an experiment can be compared to the output of the experiment. The independent of these variables is usually the input variable in any dependent random variables experiment. This definition of the dependent type of variable is by and large common throughout the world. However its application would vary a little depending on whether the experiment is statistical or is it just mathematics.
Dependent variables examples:

A medical research laboratory is studying the effect of a specific drug in treatment of cancer. Here the quantity of drug administered would be the independent-variable, and the affect the drug has on cancer would be the dependent of the variables.  This is also a statistical example of such dependent pattern variable.

The equation we saw above: y = 2x + 3 is a mathematical example. Here y is the dependent and x is the independent one. When talking of these dependent of the variables, another concept that needs to be considered is that of limited dependent variables and unlimited dependent ones.  This concept is more applicable to statistical models. There are experiments where in one independent of the variable affects more than one dependent patterns. These multiple dependent or responding variables may be limited to 2 or 4 or 10 or may be unlimited. For example if the amount of chlorine in a water supply system of a town is the responding variable, and we know that change in the chlorine amount would affect the people drinking that water, people using that water for washing clothes or utensils, the effect of such water on animals, plants, metal pipes that carry that water, etc. Therefore there are many dependent or responding variables.

Introduction of Coins as Money

Money is the only factor which every single works for and in this materialistic world one can buy anything and everything with money. Money can be termed as an essential factor for living in today’s world. There can never be a question What Coins are Worth Money since even a penny makes something complete. There are people in this world who seek for the same coin which many take it for granted and leave behind. To understand the worth of a penny one should know What is Money and the answer for that will explain and serve the purpose.

Coins are of different value according the number valued on it. Money Coins are of different size and of different value. Today, these coins are valued more, that is their value is increased earlier there has been just a low value coin and now the government has started providing coins of high value. The value of a coin can be well explained by a beggar who begs for a penny to get his / her breakfast. A coin lost will make the person understand the worth and value of it.

Coins are actually better for blind people, since it is made of steel and can be felt and sense the value of the coin just by touching it. Money Coins Pictures actually help in serving the above usage to avoid duplicate coins when given to a blind person. The best part is, a blind person can find out the money note the difference between duplicate and original.

The above is the advantage of coins, but there are disadvantages in using the coins. The coins are very small in size and it is possible for a person to lose the same out of negligence. And they make the wallet filled up and often spoil the wallet when piled up and not used. There are people who think that it is cheap if they use a coin to buy things in their daily life. The importance of coin is realized in bus when the ticket conductor has no change and we are forced to get down of the bus. The irritation increases when the same bus conductor takes some coins worth .50 or 1 rupee since he has no change to provide the passenger. The coin is good and very important to a section of people and another part take it for granted.

Conditional and Biconditional statements

A conditional statement is a statement which is performed by if true or false. For eg: if p and q are two propositions, "if p then q" is known as conditional statement or implication. A statement is called biconditional when it is expresses the idea that the presence of some property is a necessary and sufficient condition for the presence of some other property.

Conditional and Biconditional Statements:

Conditional statements and biconditional statements of different propositions may be obtained by conjunction, disjunction and negation of propositions.

Conjunction Statement:
If p and q are two propositions, then compound proposition, "p and q" is known as conjunction of the proposition. It is indicated by p q. The conjunction of two propositions p and q are true, if both p and q are true and in all other cases it is false.

Disjunction Statement:
If p and q are two propositions, then the compound proposition "p or q" is called the disjunction of p and q. It is indicated by p v q. The disjunction p v q of two proposition p and q are false if both p and q are false and in all other cases, it is true.

Negation Statement:
Let p be any proposition. The suggestion "not p" is called the negation of p. It is indicated by ~p. The negation p is false if p is true and also the negation p is true if p is false.

Example for Conditional and Biconditional Statements:

Consider the proposition, “If it is rainy then it is cloudy”, which we say is a conditional statement.
Let us consider, p =“It is raining”, q =“It is cloudy”. Then the proposition can be written as “If p then q”. We symbolize this as, p à q. This can also be deliberated as “p implies q”. We never want something false to follow from something true; i.e. we do not want “If pq” to be true if p is not true and then q is not true.

Factors of a Number

The term factor of a number is a part in which a number can be broken down. All the factors of a number when multiplied together give the number. For example number 12; let us try with a factor tree, which helps in figuring out all different factors.

To Find the Factors of a Number 12, we can see that one is divisible by 12 so one is a factor. Then 2, 3,4 are too divisible so they all are factors of 12 too. We see if 5 is a factor or not, as 5 is not divisible by 12 so 5 is not a factor of 12. Then we take 6 and see the other numbers too. Hence we can conclude that All Factors of a Number 12 are 1, 2, 3, 4, 6 and 12.

Because these are also parts that can give us 12 so these are all numbers which can be considered as Factors of Number 12. Number one is the factor of all the numbers. From the algebra perspective we can see it this way, suppose we have x^2+3x . Let’s us understand by knowing first what do they have in common. So they both have an x, hence x is considered as a factor here.  And about the left overs, x times x is x ^2 and x times what is positive 3 after taking x as common, this equation can be written as X(x+3).

So we have factors not just in numbers but also in algebraic expressions in terms of variables. Factors are basically the numbers, which a larger number can get divided by.  Similarly if we have number, its factors will be 1, 3 and 9. We can find the Missing Factor by simply dividing the factor we know by the number we have. For example: - if we have number 15 and 1 and 5 are factors given to us.

To determine its missing factors, we can divide 15 by 5 which would give us 3. Hence 3 is also a factor. Therefore 1, 3, 5 and 15 are the factors of 15. Let us do one more example of finding the factors like what are the factors of number 8. As starting with the lowest number and its partner of course one times 8 gives us 8 only. Then 2, 4 and 8 are divisible by 8 too. Thus, 1,2,4,8 are all factors for number 8.

Classifying triangles

Triangle is one of the polygon. The triangle has three sides and it falsehood in the equivalent flat surface and the calculation of internal angle measures to 180 degrees. The external angle is like to the measure of isolated interior angles in their classifying triangles. The sum of any two sides is greater the third side.

The triangles can be classifying on the basis of length of sides and the size of angles

Classifying Triangles on the Basis of Length:

The triangles can be classifying on the basis of length of sides and the size of angles.

• Equilateral Triangle,
• Isosceles Triangle,
• Scalene Triangle.

Triangles are classifying as

Equilateral Triangle:

The triangle has identical length of sides. Each angle measures to 60 degrees. It is a type of normal polygon.

Isosceles Triangle:

The triangle has two conflicting sides are parallel to each other. It also has two one and the same angles. The base angles opposite to the equal sides are equal

Scalene Triangle:

The triangle sides are unequal is normally known as scalene triangle. These types of triangles are having unequal sides

Classifying Triangles on the Basis of Angles:

Classifying triangles by using angles:

• Right angles triangle,
• Acute triangle,
• Obtuse triangle

Right Triangle:

If one angle of a triangle is a 90 degree angle, followed by the triangle is known as Right angled triangle or right triangle. Right angle is the same to 90 degrees. It follows the Pythagoras theorem.

In a right-angled triangle,

The square on the hypotenuse = sum of the squares on the legs

Acute Triangle:

When all the three angles of a triangle are acute, it is normally named as an acute triangle. Acute angle is an angle which is a slighter amount than 90 degrees.

Obtuse Triangle:

When one of the angle is obtuse in triangle, that the triangle is namely represented as an obtuse angled triangle, or an obtuse triangle. An Obtuse angle is an angle which is superior than 90  and a lesser amount of than 180 .

How to calculate Interquartile Range?

Inter quartile range is the variability measure by dividing an ordered dataset into three quartiles. It is used in the calculations of statistics instead of total ranges. Interquartile Range (IQR) is also sometimes called as the middle-fifty or midspread which measures the statistical dispersion of data.

Definition of Interquartile range

When a given set of data is divided into three quartiles based on median of those values, which will be discussed later, IQR can be defined. IQR is defined as the difference of the third or upper quartile (seventy fifth percentile) to the first or lower quartile (twenty fifth percentile) of the data in an ordered range. Half of this range of value is termed as semi-inter quartile range. The Statistics interquartile range is used to summarize the extension of data which is spread. This is considered as more effective than the median or mode values since it shows the range of dispersion rather than a single value.

Steps to Calculate the Interquartile range
Let us consider a dataset having 9 numbers 19, 20, 4, 9,8,11,15,10,12 and calculate the inter quartile range for that.
Step 1: Arrange the given set of data from smallest to largest number. Therefore the order changes to 4,8,9,10,11,12,15,19,20.
Step 2: Find the median of the series. Median is the exact middle number of a series, if the total numbers are odd. If the total number in the set is even, then the average of two middle numbers will be the median. In the above case since total numbers 9 is odd, the 5th data will be the median, which is 11.
Step 3:  Now we have to find the Q1 from left side numbers of the median and Q3 from the right side numbers of the median. From the left side 4 numbers we will get the median as (8+9)/2=8.5, which is Q1.  From the right side numbers we will get the median as (15+19)/2=17, which is Q3.
Step 4: Now, the formula for finding interquartile range is, IQR = Q3-Q1. Therefore we get the IQR as 17-8.5=8.5. Thus 8.5 is the Inter quartile range of the given series.

Thus the separation of quartile decides the value of the IQR and one should be keen on calculating the quartile values.
Alternative definition IQR can also be defined as the distance between the smallest and largest values which are present in the middle 50 percent of the dataset.  Consider a dataset having numbers 1, 3, 4,5,5,6,7,11.  After neglecting the upper and lower quartiles of the dataset, we get the remaining middle numbers as 4,5,5,6. Hence from this the IQR can be calculated as 6-4=2.

Converting Fractions to Decimals

How to Convert Fractions to Decimals
The manual method to convert Fraction to Decimal is by dividing the numerator with the denominator using the long division method for instance, 4/5 here 4 is divided by 5 using the long division method, 4 is a number less than 5 and hence we start with a decimal point in the numerator to make 4 -> 40. Now that the number is 40 using the decimal point we can divide 40 with 5 as 5 x 8 = 40, so the quotient would be 0.8 which is the decimal form of 4/5. Another method used in converting Fractions to Decimals manually is to,
• find the number which when multiplied with the denominator gives a multiple of 10.
• Once the number is found both the numerator and the denominator is multiplied with that number
• The numerator is written with an appropriate placement of the decimal point according to the multiple of 10 in the denominator
For example, 4/5 fraction to decimal first we find the number which when multiplied with 5 gives a multiple of 10. 20 times 5 is equal to 100, so the required number is 20. Now the numerator and the denominator both are multiplied with 20 which gives, 4x20/5x20=80/100, the denominator being 100 the decimal point has to be placed two places towards left from the right which gives .80 or 0.8

When we Convert Fraction to Decimal at times we might not be able to make the denominator a multiple of 10 in such cases an approximate decimal is calculated by multiplying the denominator which  gives the nearest value of multiple of 10, for instance 1/3 and 2/3 fractions to Decimals would be multiplied with 333 in the numerator and denominator which gives 333/999 and 666/999 and the decimal point is placed 3 places towards left from the right as 999 is near to 1000 which has three zeroes, so the approximate decimals are 0.333 and 0.666, the accuracy is only till three decimal places.

To convert Fraction to Decimal using a long division can be given as follows, 2/3 would be 2 divided by 3
2 is less than 3 and hence cannot be divided.  In such case a decimal point is used to make 2,  20. 20 when divided by 3 would be a repeated decimal of 6 which is written as 0.66666…. Rounding of the decimal is done if required.

                                   0.666…  (Goes on)

                                3|20

                                  - 18

                                      20

                                    - 18

                                       20

                                     - 18

                                         2  

Simplifying Fractions

Simplifying Fractions Algebra
Fractions are part of a whole number written as numerator/denominator, the numerator and the denominators are numbers that have factors other than 1 and itself, in short composite numbers. The process of simplifying fractions is a simple method of reducing fractions. Let us now learn how to simplifying fractions which leads to a reduced fraction. In simplifying fractions following are the steps to be followed:
• First we need to find the common factor of the numerator and the denominator. For instance, the common factor of 4 and 8 is 4 as 4 divides both 4 and 8 evenly.
• Next step is to divide the numerator and also the denominator with the common factor of the numerator and the denominator
• The process is to be repeated till there are no common factors for the numerator and the denominator
• Once the composite numbers of the numerator and denominator have no common factors left, the fraction is a reduced fraction or a simplified fraction

Consider the following Simplifying Fractions Examples
Simplify the fraction 48/108,
The common factor of 48 and 108 is 2, dividing 48 and 108 with 2, 48/2=24 and 108/2=54,
The common factor of 24 and 54 is 2, dividing 24 and 54 with 2, 24/2=12 and 54/2=27,
The common factor of 12 and 27 is 3, dividing 12 and 27 with 3, 12/3=4 and 27/3=9.
There are no common factors of 4 and 9 other than 1 and hence the simplified fraction is 4/9

There is another method used in Reducing Fractions or simplifying fractions, it is the GCF method. In this method, the greatest common factor of the numerator and the denominator are found. Then the numerator and the denominator are divided by the greatest common factor which gives the reduced fraction. So, in this method first the largest number that goes exactly into the numerator and the denominator is found,  9/27, here the largest number that divided 9 and 27 exactly is 9 and hence the reduced fraction would be 9x1/9x3= 1/3

Examples of Simplifying Fractions
Simplify the fraction 48/108
The greatest common factor of 48 and 108 is,
48= 2 x 2 x 2 x 2 x 3
108= 2 x 2 x 3 x 3 x 3
The greatest common factor is, 2 x 2 x 3= 12
The numerator and the denominator are divided with the greatest common factor
48/12 = 4 and 108/12=9, the simplified fraction is 4/9 which has no other common factor other than 1

Alternate Interior Angles – Properties and Examples

Definition of Alternate Interior Angles
The Alternate Interior Angles Definition states that if two lines are crossed by a transversal line, then the angles formed in the opposite side of the transversal and in the inner part of the lines at the point of intersection of the lines with the transversal line define Alternate Interior Angles.  Most of the time, the two given lines will be parallel to each other. We can also define Alternate Interior Angles as those corresponding angles which are formed in the inner side of the lines at the point of intersection of a transversal line with those lines.

Properties of Alternate Interior Angles
If the two lines crossed by the transversal are not parallel, then the alternate interior angles formed at the point of intersection of the transversal line with the parallel lines do not have any relationship with one another.  They are just alternate interior angles.

If the two lines crossed by the transversal are parallel, then the alternate interior angles formed at the point of intersection of the transversal line with the parallel lines have equal angle measure.  This means all the alternate interior angles are equal in value. Thus there exists a relationship between the alternate interior angles so formed.

Alternate Interior Angles Examples
For better understanding of the alternate interior angles, let us consider an example. Consider two lines AB and CD lying parallel to each other horizontally.  If a transversal line PQ crosses the two parallel lines, it intersects the line AB at the point E and line CD at the point F. At this point of intersection, a pair of alternate interior angles is formed in the inner part of each line and on the alternate i.e. opposite side of the transversal line.  So, totally two pair of alternate interior angles are formed.

 Among the alternate interior angles, one angle will be obtuse and the other angle will be acute.
Suppose if the alternate interior angle AEF formed in the inner side of line AB is 110 degrees, then as the lines AB and CD are parallel, the alternate interior angle EFD formed in the inner side of the line CD is also 110 degrees. This is based on the property that: in case of parallel lines, the alternate interior angles are equal.  As these two angles are obtuse, the other two alternate interior angles formed will be acute angles of measure 70 degrees (i.e. 180 - 110 = 70).  Therefore, the alternate interior angles BEF = EFC = 70 degrees.

Parallel Lines and their Properties

Parallel Lines Definition
The definition for parallel lines states that if the lines lie in the same plane and if they don’t touch or meet the other lines at any point of the line, then these lines are termed to be parallel lines. If there are two parallel-lines PQ and RS, then it is said that the line PQ is parallel to the line RS.

When are two lines said to be parallel?
Two given lines are said to be parallel-lines if it satisfies any one of the following conditions:
• If they have a pair of alternate interior angles which are of equal measure.
• If they have a pair of corresponding angles which are of equal measure.
• If any one pair of interior angles which lie on the same side of the transversal are supplementary angles.

Constructing Parallel Lines
Now let us see how to construct parallel-lines. Draw a line AB and mark a point C at some place above the line AB.  Through C, draw a transverse line which cuts AB at D. The transverse line crosses C as well as AB. The line can cut the straight line at any angle.  With D as centre and with more than half of CD as radius draw an arc which cuts CD at E and AB at F.  With C as centre and with the same radius, draw a similar arc on the transverse line above the point C to cut the transverse line at G.

Now change the radius. The width of the lower arc that crosses the two lines AB and CD is taken as the compass width i.e. the radius.  With G as centre draw an arc on the upper arc to cut at H. Now a straight line is drawn through C and H.  We can see that the straight line CH is parallel to line AB.

Thus two parallel-lines AB and CH are constructed.

Properties of Parallel Lines
Consider there are two parallel-lines A and B which are cut by a transversal line. At the point of intersection of the transversal line with the two parallel-lines, the following properties will be met:
• The pair of acute angles formed in the parallel-lines is equal.
• The pair of obtuse angles formed in the parallel-lines is equal.
• The acute angles formed are supplementary to the obtuse angles formed.
• Equally measuring alternate interior angles are present.
• Equally measuring corresponding angles are present.
• The sum of the two interior angles which are present on the same side of the transversal is equal to 180 degrees.
• The sum of any acute angle with any obtuse angle is equal to 180 degrees.

Basic understanding of reflex angles

To define reflex angle, let us look at the following figures:

What is it that is different about these angles? Or in other words, what do we notice about these angles? Note that the measures of all the angles are greater than 180 degrees. Based on this understanding we now define reflex-angles as follows: An angle whose measure is more than 180 degrees and less than 360 degrees is called a reflex angle. Mathematically that can be written like this: If an angle measure α is such that, 180 < α < 360 degrees, then α would be a reflex-angle measure.

Now that we know what a reflex-angle is, the next most obvious question would be how can a reflex-angle be measured? Considering the fact that normally we use a set square or a protractor to measure angles, we know that the maximum angle that can be measured using a protractor or a set square is 180 degrees. So how can we measure reflex-angles?

How to measure a reflex angle?
Every normal angle, which is not a reflex-angle, has its corresponding reflex-angle.  Whether an angle is acute or obtuse, it would always have its corresponding reflex-angle. This can be seen in the examples below:
Example 1: 

Here the acute angle is 49 degrees and the corresponding reflex-angle is 311 degrees. The sum of these two angles is 49 + 311 = 360 degrees.
Example 2: 

Here we have an obtuse angle measuring 112 degrees and its corresponding reflex-angle measuring 248 degrees. The sum of these two angles is 112 + 248 = 360 degrees.

Therefore based on the definition of reflex angle, we can state that the sum of an angle and its corresponding reflex-angle is always 360 degrees. Thus if we want to measure a reflex-angle, we follow the following steps:
1. First we measure the corresponding acute or obtuse angle (say 112 degrees or 49 degrees in the above figures) = α degrees.
2. Now subtract the angle thus measured from 360 degrees. Thus our reflex-angle
= r = 360 - α degrees.

Where do we find reflex angles?

Reflex-angles are usually found in concave polygons.

A concave polygon would have at least one reflex-angle. Other examples of concave polygons are shown below:

Here we have a concave quadrilateral, a concave pentagon and a concave heptagon.

Geometry: Alternate interior angles

Definition: Alternate interior angles
Consider a pair of parallel lines is intercepted by a transversal. At each of the intersection points of the lines with the transversal, 4 angles are formed, making a total of 8 angles for the two lines. Each of these angles have a name or significance. Let us try to understand the following example of alternate interior angles.

The above figure shows two black lines intercepted by the red transversal. In the interior of the lines, four angles are formed namely, <5 a="a" also="also" alternate="alternate" and="and" angles.="angles." angles="angles" are="are" called="called" congruent.="congruent." green="green" if="if" interior="interior" is="is" of="of" other="other" p="p" pair="pair" similarly="similarly" the="the" these="these">
Theorem related to alternate interior angles:
When a pair of parallel lines is intercepted by a transversal, each pair of alternate interior angles thus formed are congruent. Therefore in the above figure, angle <5 and="and" angle="angle" congruent="congruent" is="is" p="p" to="to">
Examples of alternate interior angles:
The following pictures show examples of alternate interior angles:
Example 1:

In the above figure, the angles 76 and b are alternate interior angles. Therefore we can say that measure of angle b is 76. Similarly measure of angle a would be 104 since these two are also alternate interior angles and we know that alternate interior angles are congruent.

If one pair of alternate interior angles is acute, then the other pair of alternate interior angles has to be obtuse.  (Note an acute angle is an angle whose measure is less than 90 degrees and an obtuse angle is an angle whose measure is more than 90 degrees but less than 180 degrees)
Example 2:

In the above figure, the parallel lines are intercepted by a horizontal transversal. So here the purple dots are a pair of obtuse alternate interior angles and the pink dots are a pair of acute alternate interior angles. As we already know both the purple angles have to be congruent to each other and similarly both the pink angles also have to be congruent to each other.

Introduction to concept of median

What is median?
Median in math has two meanings. One is the geometric meaning and another is the statistical median.

Geometric median definition:
In a closed plane figure such as a triangle, the line segment that connects the midpoint of one side to the opposite vertex is called the median. See picture below:

The above picture shows a triangle ABC. D, E and F present on the sides AB, BC and CA, so that, AD = DB, CE = EB and AF = FC. The line segments AE, CD and BF are from the triangle ABC. The points where all the three geometric mid-segments intersect is called the centroid of the triangle also called the centre of gravity. In the above figure, O is the centroid of the triangle.

Statistical median definition:

In statistics it is a measure of central tendency. In a frequency distribution, the central value around which most of the values of the variable are centered is called the measure of central tendency. Of the various measures of central tendencies, the most popular are mean, middle number and mode. It is defined as middle value of the data set. For finding median we need to follow the following steps:

1. Arrange the data set in ascending or descending order.
2. If the number of entries is odd, then the middle value would be the (n+1/2)th value.
3. If the number of entries is even, then the middle value would be the average of the (n/2) th and the (n/2 + 1)th value.
4. The value found in step 2 or 3 is called the middle number value.

Sample problem:
1. Find the middle number of the following data set of marks obtained by 10 students in a class test of maximum 10 marks: 8, 8, 9, 5, 5, 6, 6, 7, 6, 4
Solution:
Step 1: Arrange the data in ascending order: 4, 5, 5, 6, 6, 6, 7, 8, 8, 9
Step 2: The number of entries is 10 which is an even number, so we move to step 3.
Step 3: The two middle values would be 10/2 = 5th and 10/2 + 1 = 5+1 = 6th value. Both these are 6. Therefore the middle number is 6.

Elementary math concepts

Elementary math covers all the basic operations, function and concepts in algebra. In elementary math is the main part covered is arithmetic operations. All the basic operations are presented in this area. This is covering all basic operation of addition multiplication, subtraction, and division. The elementary algebra covers all concepts from kindergarten level to middle school level. This is also help to solve for real life math problem.

Example: 4x+ 2 = 1

Elementary Math Concepts Covers:

Elementary math concepts contain this basic operations

Arithmetic Operations:

The real numbers have the following properties:

a + b= b +  a    ab  = ba                            (Commutative Law)

(a+ b)+ c= a+ (b + c)      (ab)c = a(bc)        (Associative Law)

a+(b +c)= ab +ac                                       (Distributive law)

Fractions:

To add two fractions numbers with the same denominator, we use the Distributive Law property:

a/b+c/b= 1/b*a+1/b*c  =1/b(a+c)  =a+c/b

To add two fraction with different denominators, we use a frequent denominator:

a/b+c/d =  ad+bc/bd

Factoring

Here we can make use of Distributive Law to increase certain algebraic conditions. In rare case we need to repeal this method (again using the Distributive Law) by factoring an expression as a product of simpler ones. The easiest condition occurs when the given expression has a common factor as given below...

3x(x-2)=3x2 – 6

Elementary Algebra Concepts Problems

Q 1 :   Find the larger number, -13 or -16?

Sol :  If a number has a negative sign, we dont conclude the number by ony seeing the value, here we have to consider the negative sign. If the value of a number with negative sign increases, the actual value of the number is decreses.

-16,-15,,-14,-13,-12,-111,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0

so,-13 is larger than -16.

Q 2:  List all the integers between -2 and 4.

Sol :     -2,-1, 0, 1,2,3,4 these number are present in the -2 ,4

The -2, 4 between numbers are -1,0,1,2,3

Q 3:  Reduce 14/35.?

Sol :      14/35

We divide 7 on both sides

2/5

Q 4: Simplify 8 -: 2//3?

Sol :  We divide the given equation this is simple method the division inverse of multiplication

8

4*3   = 12

Q 5:  Simplify 1/2 + 2/3?

Sol  :      1/2+ 2/3

We take L.C. M   on 2, 3

1*3/2*3 +2*2/3*2

=3/6+4/6 =3+4/6

=7/6

Q 6.12x=4?

Sol :    x=4/12

x=1/3

Q 7:    3X+4y=5x-2y ?

Sol :         4Y+2y=5x-3x

6y=2x

y=2x/6

y=x/3

Q 8 :  x-4=8?

Sol : We add +4 on both sides

x=8+4

x=12

Q 9:  Factorize the given expression x2-9   ?\

Sol :  The general form of

(a2-b2)=(a-b)(a+b)

So, x2-9=  (x-3)(x+3)

Adjacent Angles

Definition for Adjacent Angles states that two angles are said to be adjacent, if:
• Both the angles are formed using the same side
• Both the angles have a same corner point i.e. the vertex
• Both the angles do not overlap on each other i.e. they should not have any interior point in common.

In simple words, Adjacent Angles Definition states that angles that are formed side by side using a common ray coming out of a common vertex in such a way that the common ray is between two other rays that forms the angles without any overlapping.

If two angles are given, they are said to be adjacent if they are only as per the Definition Adjacent Angles has stated.  There are certain scenarios when two angles do not satisfy the conditions stated in the Adjacent Angles Definition, they are:
1. Two angles share a common corner point or vertex but do not share a common side.
2. Two angles share the same side to form the angles, but do not have a common point at one of its corners.
3. Two angles given, in which one angle overlap the other.
The above cases that do not satisfy the conditions stated in the definition of adjacent angles can be declared as not adjacent. So these angles are not adjacent to each other.

Example of Adjacent Angles
An example of Adjacent-Angles helps to understand the concept in a better way. Consider three rays A, B, C coming out of the common vertex O. Two angles namely Angle x and Angle y are formed in such a way that angle x is formed between the sides OA and OB whereas angle y is formed using the sides OB and OC.  Here the vertex O is used as common and the side OB is used in common to form both the angles.

Adjacent Angles as Complementary Angles
When there are two adjacent-angles given with common vertex and common side, find the sum of the two angles.  If the total of the two angles is ninety (90) degrees and if it forms a right angle, then these adjacent-angles are said to be complementary and are termed as complementary angles. We can call it as adjacent complementary angles too.

Adjacent angles as Supplementary Angles
If two adjacent-angles are given, we can say that these adjacent-angles are supplementary angles if the total of the two adjacent-angles given is hundred and eighty (180) degrees and forms a straight angle. We can call it as adjacent supplementary angles.

Hypotenuse of a Right Triangle in brief

There are different types of triangle which we have learnt; one of them is the right angled triangle in other words a right triangle. A triangle in which one of the angles is a right angle that is 90 degrees is called a right triangle. The longest side of a right triangle is called the Hypotenuse of a Right Triangle and the other two sides are called the legs of the right triangle. We use the Pythagorean Theorem in finding the hypotenuse of a Right Triangle. The Pythagorean Theorem states that ‘the sum of the squares of the two sides (legs) of a right triangle is equal to the square of the hypotenuse’. Let us assume the lengths of the legs of a right triangle to be ‘a’ and ‘b’ units and the hypotenuse length to be ‘c’. By using Pythagorean Theorem, we can calculate the hypotenuse of a right triangle,
(Hypotenuse) ^2 = (sum of the squares of the sides (legs)^2
  c^2 = (a^2 +b^2)

This gives us the Hypotenuse of a Right Triangle formula, c = sqrt(a^2 + b^2)
Given the lengths of the sides or legs of a right triangle as 3 cm and 4 cm respectively, find the hypotenuse of the right triangle.  Here we are given the lengths of the two sides a = 3cm and b = 4cm, we need to find c. let us apply the Pythagorean Theorem, we get, c = sqrt(a^2+b^2). a^2= (3)^2 = 9 and b^2 = (4)^2 = 16, that gives us a^2+b^2 = 9 +16 = 25. So  c = sqrt(25) = +/- 5 , as length cannot be negative, the hypotenuse of the given right triangle is 5 cm.

Let us now learn how to calculate the Hypotenuse of a Right Triangle using the Pythagorean theorem given by hypotenuse = sqrt[(sum of the squares of the sides or legs)^2]. A ladder is placed against a wall of height 12 ft. The distance between the base of the ladder and the wall is 5ft, find the length of the ladder. In this problem, the triangle formed by the wall, the floor and the ladder is a right triangle and hence the length of the ladder would be the hypotenuse which we need to find. We are given the two lengths of the sides of the triangle which are 12ft and 5 ft respectively. We know c= sqrt(a^2 +b^2); here a = 12ft, b = 5 ft which gives us a^2 = 144 and b^2 = 25. So, c = sqrt(144 +25) = sqrt(169) = +/-13. The length of the ladder is 13 ft

Solving fourth grade math homework

In this article we are going to discuss about the mathematical concepts for fourth grade students. The students of fourth grade learn the different areas of mathematics, like place values of six digits numbers, expanded notations, place value chart, addition, subtraction, multiplication and division of three and four digits numbers, multiples and factors, HCF and LCM. Grade fourth students also learn unitary method, fractions, decimal numbers and measurement of time, length, mass and capacity.

Here we are going to discuss about some of them. This article will be helpful in solving fourth grade math home work.

Topics of Solving Fourth Grade Math

Some of fourth grade mathematical topics are as follows:

Place value: To read and write large number easily, the Indian place value chart is divided into periods as shown below:

Practice Questions: 

(1)Write the number name of these numbers:

(a)    2, 50,946 = Two Lakh fifty thousand nine hundred forty six.

(b)    6, 92,438 = Six lakh ninety two thousand four hundred thirty eight.

(c)    20, 10,101 = Twenty lakh ten thousand one hundred one.

 (1)   Write the numeral of these number names:

(a)    One lakh fifty thousand two hundred eighteen = 1,50,218

(b)   Nine lakh ninety five thousand sixty three = 9,95,063

(c)    Seventeen lakh fifteen = 17,00,015

Simplification involving four fundamental operations: 

In this lesson we will learn to use all the operations together. The fourth grade learners are able to learn the order of operations through this section.

Step 1 ---- Of

Step 2------Division

Step 3 ------ Multiplication

Step 4 ------ Addition

Step 5 ------ Subtraction

 So the order of operation is ODMAS.Now we will do some simplification using ODMAS rule:

 Practice Questions:

(a)    Simplify 36÷ 6 x 4 + 2 – 8

                   = 6x 4 + 2 – 8

                   =   24 +2 – 8

                    =    26 – 8 =18 Ans.

(b)   Simplify 6 + 8 ÷2 -2 x 1 + 5 of  2÷ 5

                    = 6 + 8 ÷ 2 – 2 x1 + 10 ÷5

                    = 6 + 4 – 2 + 2

                    = 12 - 2 = 10 Ans.

Now simplify these questions:

(a)    23741 -  3826 ÷  2 x 6 + 221

(b)   529 x 71 – 630 of 6 ÷ 3 + 4

(c)    6000 +9000 ÷  500 of 6 – 2000

(d)   6699 ÷ 33 +3075 ÷ 25 – 203

Answers: (a) 12484 (b) 36303 (c) 4003 (d) 123

Solving Fourth Grade Math Homework-unitary Method

Some basic knowledge is very important for fourth grade learners like:

• We have to add when we find the total cost of things tat we purchase.
• We have to subtract when we take back the balance.
• We have to multiply to calculate the cost of more articles.
• We have to divide when we find the cost of one.

Practice Question: 

If the cost of 25 Milton jugs is Rs. 8,250.  What is the cost of 48 such Milton jugs?

Solution: The cost of 25 Milton jugs = Rs.8, 250

                The cost of 1 Milton jug is Rs.8, 250 ÷ 25 = Rs.330

                The cost of 48 Milton jugs = Rs. 330 x 48 = Rs.15840

 Solve these problems: 

 28 digital diaries cost Rs.70, 560. What is the cost of 15 such diaries?

 Answer: 37,800

Statistics homework answers

Statistics is the branch of applied mathematics which deals with scientific analysis of data. The subject statistics had been started in early days as arithmetic to aid a ruler who needed to know the wealth of his subjects to levy new taxes. Now a days statistics  plays an important role in all organizations in their decision-making and planning. Statistics homework deals with mean,median and mode.

Classification of Statistics Homework Answers:

In statistics homework answers has following topics are

• Arithmetic mean
• Median
• Mode

The arithmetic mean;

In Statistics the arithmetic mean (A.M) or simply the mean or average of n observations x1, x2, …, xn is defined to be number x such that the sum of the deviations of the observations from x is 0.

         x1 +x2 +x3.......xn
x =  --------------------
                     n

the symbol Σ, called sigma notation is used to represent summation.

x=Σ xi
   -------
      n

Homework Example:

Calculate  mean of the data 9, 11, 13, 15, 17, 19.

Sol :

X =   Σxi/n = `[9+11+13+15+17+19]/[6] ` =` [14]/[6]`

Median :

Median is defined as central most or middle value for given series of data it should be arranged in ascending or descending order.

Homework Example:

Find the median of 23, 25, 29, 30, 39.

Sol:

The given values are already in  ascending order. No. of observations N = 5.

So the median = ` (N+1)/(2)`  =`(5+1)/(2)`
= 3 rd term =29
∴ Median = 29.

Mode:

In Statistics answers,Mode is also a measure of central tendency.

In a set of each observations, mode is defined as value which occurs most often.

If the data are arranged in the form of a frequency table, the class corresponding to  greatest frequency is called  modal class.

Homework Example:

Find the mode of 7, 4, 5, 1, 7, 3, 4, 6,7.

Sol:

Arranging the data in  ascending order, we get 1, 3, 4, 4, 5, 6, 7, 7, 7.

In above data 7 occurs several times. Hence mode = 7.

Homework Problems and Answers for Statistics:

Calculate mean :    
Problem 1:
Calculate the mean of the data 7, 12, 18, 14, 19, 20.
Problem 2:
Calculate the mean of the data   16,18,14,15,21,30,26,44
Problem 3: 
Calculate the mean of the data   22,77,55,11,22,26,38,72
Answers:
1) 15
2) 23
3) 40.35

Calculate median:
Problem 1: 
Find the median of 3, 4, 10, 12, 27, 32, 41, 49, 50, 55, 60, 63, 71, 75, 80.
Problem 2:
Find the median of 29, 23, 25, 29, 30, 25, 28.
Problem 3: 
Find the median of 26, 25, 29, 23, 25, 29, 30, 25, 28, 30.
Answers:
1) 49
2) 28
3) 27

Calculate mode:
Problem 1:
Find the mode of 12, 15, 11, 12, 19, 15, 24, 27, 20, 12, 19, 15
Problem 2:
Find the mode of 3,5,8,3,9,3
Problem 3: 
Find the mode of 3,5,8,5,6,7
Answers:
1) 12 and 15
2) 3
3) 5
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An Introduction to Calculating the Area of a Right Triangle

The Area of a Right Triangle
A Right Triangle or a Right Angled Triangle is a triangle which has right angle or 90 degrees as one of its interior angles. We have two types of Right Triangles namely, scalene right triangle in which one angle is right angle and the other two are unequal angles and the corresponding sides are also unequal. The other type is the Isosceles Right Triangle in which one angle is right angle and the other two angles are equal and also their corresponding sides are of equal length.  A right triangle is very useful in trigonometry, a branch of mathematics. Area of a triangle is given by half times the base times the height of the triangle. We can deduce the formula for Area of a Right Triangle using the following method. Let us consider a rectangle of length l cm and width w cm. Let us now cut the rectangle into two equal halves diagonally as shown in the figure below.

 
The triangles ABC and ADC are congruent triangles which means when placed one over the other they overlap exactly. Also they are right triangles with 90 degrees as one of their angles. They are of the same size and hence we can say that they have the equal area. From this it is clear that the area of a right triangle will be half the area of the rectangle.
The Area of the right triangle = ½ [Area of the Rectangle]
               = ½ [length x width]
             = ½ [l w]

From the figure, we can see that the length of the rectangle is one of the sides of the triangle which is called the base of the triangle. So, we have base of the triangle equals length of the rectangle.  Also the distance from the vertex A of the triangle ABC to the vertex B is the height of the triangle ABC which is the width of the rectangle. So, we have height of the triangle equals width of the rectangle. Substituting these in the above equation, Formula for Area of a Right Triangle = ½ [base x height]. So, to calculate area of a right triangle we use the formula ½ times [base x height]
Calculate the Area of a Right Triangle with base and height 12 cm and 17 cm respectively
Solution:  Base of the right triangle   = 12 cm
         Height of the right triangle = 17 cm
       Area of the right triangle   = ½ x base x height
                      = ½ x 12 x 17 = 6 x 17 = 102 cm^2

Characteristics of Right Angle Triangle

Triangle is a basic shape in geometry. A right triangle means 90 - degree angle that is one angle is right angle. It also called as right angled. It shows the relations between the sides and angles of a right triangle and it is the basic concept of trigonometry. The opposite side of the right angle is known as hypotenuse. The adjacent side to the right angle is known as legs or sometime known as catheters. This phenomenon is the basis of Pythagorean triangle theorem. This theorem states that the length of all three sides of a right triangle are integer and the side length are known as Pythagorean triple.

The basic principle properties of a right triangle are area, altitude, Pythagorean Theorem, inradius and circumradius. First we discuss about area of a right triangle. Area of a right triangle is equal to the one half of the base multiplied by the corresponding height. It is true for all right angled triangles. In a right triangle from the two legs if one leg is taken as a base then other leg is taken as height. So in other way we can say that area of a right triangle is one half of the product of the two legs means one half products of base and height. Mathematically suppose in a right triangle two legs are (a) and (b) and hypotenuse is (c) then formula for the area of a right triangle is expressed as [A=(1/2 ab)].

For finding area of a right triangle we have to calculate perimeter also. The formula we can see above. Suppose in a right triangle two legs (a=5) and (b=12) where a is base and b is height. Hypotenuse (c=13) is given. We know the formula area of a right triangle so we substitute all the values in formula. Finally we get area= (1/2 *5*12) = (1/2*60) =30 and unit of right angle triangle is square meter or square centimeters.

Similarly we can find surface area of a right triangle.  Formula for surface area of right angle triangle is [Area (A) =bc/2] where (a=√b^2+c^2), (b=√a^2-c^2) and (c=√a^2-b^2). Suppose for above problem we have to find surface area. We substitute all the values in formula that is (A=12*13/2).
From the right angle triangles we also find the Pythagorean Theorem. It is very useful in all parts of mathematics. It states that in any right angle square of hypotenuse is equal to sum of square of base and square of height that is (c^2=a^2+b^2).

Trigonometric equations

Trigonometric equations
In this article we shall study some basics of how to solve trigonometric equations. A trigonometric equation is an equation that involves trigonometric ratios in addition to algebraic terms. As trigonometric  functions are many to one type of functions, it is quite possible that such trigonometric- equations may have many roots (solutions).

This can be made clear by solving trigonometric equations examples:
For instance, sin ( pi/6) = ½ but the equation sin x = ½ has not only the solution x =  pi/6, but also x = 5 pi/6, x = 2 pi +  pi/6, x = 3 pi -  pi/6 etc. Thus, we can say that x =  pi/6 is a solution of sin x = ½ but it is not the general solution of the equation.

A general solution gives all the roots of an equation. When attempting to solve trigonometric equations online it is very imperative that the equations fed in are valid. For example, and equation like sin x = 2 would not have any solution. That is because we know that the range of the sine function is [-1,1]. So for any value of angle x, the value of sin x can never be = 2. Therefore the solution of such an equation would be : no solution.

The objective should be to develop methods to find general solutions of trigonometric-equations.
We know that sine, cosine and tangent functions are all periodic. Sine and cosine functions have a period of 2 pi and the tangent function has a period of  pi. Therefore the general solutions of the equations, sin t = 0, Cos t = 0 and tan t = 0 can be found as follows:
Sin t = 0 ? t = k pi, k belongs to the set of integers ------------- (1)
Cos t = 0 ? t = (2k+1)/2, k belongs to set of integers ----------- (2)
Tan t = 0 ? t = k pi, k belongs to the set of integers -------------- (3)

These results can be used to solve the general trigonometric-equations which are as follows:
Cos t = a, |a| = 1
Sin t = a, |a| = 1
Tan t = a, a’ belongs to set of all real numbers.
By solving a trigonometric equation we intend to find a set of solutions for the given set of trigonometric-equations such that each member of the solution set satisfies the set of equations.

Learn more about how to solve Trigonometry Problems.

Limit of a function

For understanding the limit of a function we must know the fundamental concepts of calculus and analysis which belonging to that particular function near a define input. In another way we can simplify by taking a simple example. Suppose function is f(x) and limit is x tends to c where c is a constant, then it means that function f(x) is get closer to limit as x get closer to c. more accurately when function is applied to each input , the result in an output value that is close to limit.

Limits of a sequence
As we know about the limits. Limits of a sequence means a value in term of sequences. If limit value exist then such sequence occurs. Limits of a sequence of any function can be understood by applying the function on a real line. This is one way to know the limits of function in term of sequence.

Limits of sequences
For knowing the limits of sequences we choose such type of example where limits value exist then the sequences will convergence. Limits are first apply for the real numbers and then for others such as metric spaces and topological spaces.

Limits of functions
In limits of functions we take any calculus function like additive or subtractive functions, because for simplify purpose these functions are easy. Functions are with limits where in limit x is approaches to any real number. When we solve such function in last we must put the limits. So in limits of functions output result are moves according to limits.

Limit of sequence
In limit of sequence, first take a sequence such as (Xn) with limit where x is tens to a and a is a constant. When we apply limit to this function if and only if for all sequences(Xn) means with (Xn) with all value of n but not equal to constant a.

Limits of a function
Limits of function means limits should apply to a function. Function may be different types. In limits of function we explain that what it means for any function which tends to real number, infinity or minus infinity. Limits of function can be both right handed and left handed.
Suppose in any case if both right hand and left hand limits of  function as x approaches to constant exist and are equal in value, their common value evidently will be the limits of a function. If however either or both of these limits do not exist then the limits of a function does not exist.

Triple integral solver

We know that when solving double integrals, we divide the two dimensional region into very small rectangles. Then we multiply the area of the rectangles with the value of the function at that point. Then we sum up these areas and then apply the following limit: lim (size of rectangle -> 0). Doing all that gives us the double integral of the said function.

Let us now try extending this concept to three dimensions. Just like in double integrals we had some region in a plane (say the xy plane), in triple integral we will consider a solid in space (i.e. xyz space). Just like in double integrals we had split the region into rectangles, in solving triple integrals we break down the solid into numerous rectangular solids (or cuboids). Extending further on same lines, just like how in double integrals we multiplied the value of the function by the area of the respective rectangle, in a triple integral example, we would multiply the value of the function at each of the point by the volume of the rectangular solid at that point. Instead of the limit of size of rectangle tending to zero, in case of triple integrals we have the limit as the volume of rectangle tending to zero.

With that we come to the definition of a triple integral. Which is like this: Triple integral is defined as the limit of the sum of product of volume of rectangular solids with the value of the function.
We call the double integral as an equivalent to double iterated integral. In the same way we can understand the triple integral as a triple iterated integral.

Symbolically the definition of a triple integral can be stated as follows:
Consider a function f(x,y,z). It is of three variables. It is continuous over any solid S. Then the triple integral of this function over the solid S can be symbolically stated as:

where the sum is taken over all the rectangular solids that are contained in the solid S. The limit is for the side length of the rectangles.
The above definition of triple integrals is useful only when we are given a set of data in the form of a table of values of volume and value of function. When a function is defined symbolically, then we use the Fubini’s theorem to solve triple integral examples.

First and Second Fundamental theorem of calculus

First Fundamental Theorem of Calculus: If f is continuous on [a, b], then F(x) = integral a to x f(t) dt has a derivative at every point of [a, b] and dF/dx = d/dx integral a to x f(t) dt = f(x), a is less than and equal to x is less than equal to b. Let us understand Proof of Fundamental Theorem of Calculus: We prove the fundamental theorem of calculus by applying the definition of derivative directly to the function F(x).

This means writing out the difference quotient F(x + h) – F(x) /h and showing that its limits as h approaches to 0 is the number f(x). When we replace F(x + h) and F(x) by their definite integrals, the numerator in above equation becomes F(x + h) – F(x) = integral a to x+h f(t) dt – integral a to x f(t) dt. The additive rule for integrals simplifies the right hand side to Integral x to x+h f(t) dt So that the above equation becomes  F(x + h) – F(x) /h = 1/h [F(x + h) – F(x)] = 1/h integral x to x+h f(t) dt. According to the mean value theorem for definite integrals, the value of the last expression in the above equation is one of the values taken on by f in the interval joining x and x + h. That is for some number c in this interval, 1/h integral x to x + h f(t) dt = f(c).

We can therefore find out what happens to (1/h) times the integral as h approaches to 0 by watching what happens to f(c) as h approaches to 0. As h approaches to 0, the endpoint x + h approaches x, pushing c ahead of it like a bead on a wire. So c approaches x, and since f is continuous at x, f(c) approaches f(x) .Lim h approaches 0 f(c) = f(x).Going back to the beginning, then we have dF/dx = lim h approaches to 0 [F(x+h) – F(x)] /h= lim h approaches 0 1/h integral x to x+h f(t) dt = lim h approaches 0 f(c)= f(x). This concludes the proof.

Let us more understand this through Fundamental Theorem of Calculus problems: let us take few fundamental theorem of calculus examples .suppose we have Find dy/dx if y = integral 1 to x^2 cos t dt.

Now to understand this solution suppose let us notice that the upper limit of integration is not x but x^2. To find dy/dx we must therefore treat y as the composite of y = integral 1 to u cos t dt and u = x^2 and apply the chain rule: dy/dx = (dy/du).(du/dx) = d/du integral 1 to u cos t dt . du/dx = cos u . du/dx= cos x^2 . 2x = 2x cos x^2.

Second Fundamental Theorem of Calculus: If f is continuous at every point of [a, b] and F is any anti-derivative of f on [a, b], then Integral a to b f(X) dx = F(b) – F(a). Let us understand this by second Fundamental Theorem of Calculus Examples suppose we have Evaluate integral 0 to pi cos x dx. Now let us solve this integral 0 to pi cos x dx = sin x 0 to pi = sin pi – sin 0 = 0 – 0 = 0.

Derivative of Logarithmic Functions

Derivative of Log Function
An exponential function is given by y=b^x, an inverse of an exponential function is given by x=b^y. The logarithmic function with base b is the function given by y=log x with base b where b>1 and the function is defined for all x greater than zero. The system of natural logarithm has a number e as its base. We know that the natural logarithms functions and natural exponential functions are inverses. If f(x) and g(x) are inverses of each other then the derivatives of inverses is given by g’(x)=1/f’[g(x)].So, the Derivative of Log Function can be found using the definition of derivatives of inverses as follows, given
f(x)=e^x and g(x) = ln x then, g’(x) = 1/f’[g(x)]
                                     =1/e^[g(x)]
                     = 1/e^ln(x)                 [g(x)=ln x]
                     = 1/x
So, the Derivative of Log Function is, d/dx[ln x]=1/x where x is greater than zero

Derivative of Log Base a of X 
We can find the Derivative of the general logarithmic function by the method of change of base formula.
 Log base a of x  can be written as ln x/ln a.
 Differentiating on both sides gives,
d/dx [log base a of x]= d/dx [ln x/ln a]
                                        =1/ln a . d/dx [ln x]
                                = 1/x ln a
Derivative of Log Base a of X is given by, d/dx[log base a of x] =1/[x ln a]
For example, find the derivative of log base 2 of x .
By the base change formula, we get, log base 2 of x = ln x/ln 2
Taking the derivative on both sides, d/dx [log base 2 of x] = d/dx [ln x/ln 2]
Here, ln 2 is a constant. On simplifying, we get, 1/ln 2 . d/dx [ln x]  
We know that derivative of ln x is 1/x. Substituting this value, we get
                        d/dx[log base 2 of x] = 1/[x ln 2]

Derivative of Log Base 10
Logarithms to base 10 are called the common logarithms. Common Logarithm is written as log x which means log base 10 of x.  Given y=ln x its inverse is given by e^y = x. Let us find the Derivative of Log Base 10 by taking derivatives on both sides with respect to x, d/dx [e^y] = d/dx[x] which gives  e^y dy/dx = 1. We have e^y= x, substituting this value, we get x dy/dx=1. We need dy/dx, so we shall divide on both sides with x which gives, dy/dx = 1/x. Derivative of Log Base 10 is given by, d/dx [log base 10 of x] = 1/x

Inverse Trigonometric Functions

Consider a function f(x)=(x+2), we can write this as y=x+2. Solving for x, we get, x = y-2. Interchanging the x and y terms gives a new function y=x-2, consider this as g(x). This new function g(x) is the inverse function of f(x). Inverse function is denoted as f^-1

Inverse Trigonometric Functions
Let us assume a given radian angle pi/2, we can evaluate a trigonometric function sin(pi/2)  equals 1. Inversely, if we are given 1 as the value of a sine function, then we can arrive to the radian angle y. Equating,  sin(pi/2) = 1, the radian angle for which the value of sine is 1 is pi/2. We can now write the inverse trigonometric function of sine as arc[sin(1)]= pi/2. In general, given the value x of the trigonometric function we can write the inverse trigonometric function as arc[sin(x)], arc[cos(x)],arc[tan(x)], arc[csc(x)], arc[sec(x)], arc[cot(x)] to name the radian angle that has the value x in each case.

Using the derivatives of inverse trigonometric identities we can obtain inverse trig functions integrals:

Integral[du/sqrt(a^2-u^2)] = sin^-1[u/a] +c
Integral[du/(a^2+u^2)=1/a[tan^-1(u/a)]+c
Integral[du/u[sqrt(u^2-a^2)]=1/a[sec^-1mod(u/a)] +c;
where u is a function of x, which is written as u=f(x)

How to Solve Inverse Trig Functions
Steps involved to solve inverse trig function, arc tan(-1):
Step1:  let y=arc tan(-1)
Step2:  tan (y) is -1 when pi/2<=y<=pi/2 [finding the interval]
Step3:  determine the radian angle of tan that gives the value of 1, which is pi/4
Step4:  determine the radian angle for a negative value, we know that tan(-x)=- tan(x)
Step5:  we get, tan y = -1
Step6:  the required radian angle is y = pi/4

Inverse Trig Functions Problems
Evaluate arc cos(-1/2)
Let y = arc cos(-1/2), we need to find the radian angle y
cos(y) = -1/2 , we know that pi/3 radian angle of cos gives the value 1/2. But here we have -1/2
We know that, cos(pi-x) gives us –cosx.
So, cos(pi-pi/3) = -1/2
We get, y=pi – pi/3 = 2pi/3 is the required radian angle

How to Graph Inverse Trig Functions
To graph inverse trig functions, first we need to understand the domain and range of the inverse trig functions which are:
Function Domain Range
y=arc sin(x)  -1<= x<= 1    -(pi/2)<=y<=(pi/2)
y=arcos(x)   -1<= x<=1   0<=y<=pi
y=arctan(x)   -infinity
y=arccsc(x)  x<=-1or x>=1  -pi/2<=y<=pi/2, y not equals zero
y=arcsec(x)   x<=-1 or x>=1  0<=y<=pi, y not equals pi/2
y=arccot(x)      -infinity  0

Plug in different values of x (within the range) to arrive to the respective y values of the given inverse trig function. Plot the coordinates (x,y) on the graph and join them. The curves thus obtained is the graph of given inverse trig function.