Exterior angles of polygons

Definition:
For any polygon, the angle formed between any one side of the polygon extended and the next consecutive side is called an exterior angle of that polygon.  See the figure below:

The sum of an internal angle and its corresponding exterior angle is always 180 degrees. In other words the interior angle and the exterior angle of any polygon are supplementary to each other.

A polygon has as many numbers of exterior angles as interior angles. The following figure shows the exterior angles of a pentagon.

Exterior angles theorem:

The exterior angles theorem can be stated as follows:
“The sum of exterior angles of a polygon is always 360 degrees. “
The sum of exterior angles does not depend on the number of sides of the polygon. The sum of interior angles would be different for different polygons. But the sum of exterior angles of a polygon is always 360 degrees.

Proof of the above theorem:

We know that the sum of the interior angles of a triangle = 180 degrees
The sum of interior angles of a quadrilateral = 360 degrees
The sum of the interior angles of a pentagon = 540 degrees
Therefore the sum of interior angles of a polygon of n sides = (n-2)*180 degrees
Therefore the measure of each interior angle = (n-2)*180/n degrees
Since each interior and its corresponding exterior angles are supplements of each other,
Therefore the measure of corresponding exterior angle = 180 - (n-2)*180/n
Therefore the sum of n such exterior angles
= n*[180 - (n-2)*180/n]
= n*[180n – 180n + 360]/n
= 180n – 180n + 360
= 360 degrees
Hence it is proved that irrespective of the number of sides of the polygon, the sum of exterior angles is always 360 degrees.

Exterior angles of a polygon formula:
The formula for the sum of interior angles of a polygon is
= (n-2)*180 degrees

Therefore each of the interior angle would be
= (n-2)*180/n degrees

Therefore the corresponding exterior angle for that polygon would be:
= 180 - (n-2)*180/n degrees

Thus the formula for finding the measure of each exterior angle of an n sided regular polygon is
180 - (n-2)*180/n degrees

Frequency table math

In statistics frequency table refers to the tabular representation of frequencies of a sample. The frequency distribution of a sample can be represented in three ways: (a) Textual representation, (b) tabular representation and (c) diagrammatic representation.

(a) Textual representation: This method comprises presenting data with the help of a paragraph or a number of paragraphs. The official report of an enquiry commission is usually made by textual presentation. The merit of this mode lies in its simplicity and even a layman can present data by this method. This method is however not preferred by statisticians simply because it is dull monotonous and comparison between different observations is not possible in this method.

(b) Tabular representation: We can define frequency table as a systematic presentation of data with the help of a statistical table having a number of rows and columns and complete with reference number, title, description of rows as well as columns and foot notes, if any. This method is any day better than the textual representation because it facilitates comparison between rows and columns. Complicated data can also be represented using a frequency table. To be able to make a diagram for the distribution we first need the data in tabular form. Only then can we convert it to a bar chart or a pie chart. All other parameters such as mean, median, mode, standard deviation, regression analysis etc are not possible unless we have the frequency distribution table.

Example of a frequency table:
The production of rice in a particular region for each year starting from the year 2001 to the year 2007 is given in the frequency table below:

Year	Quantity in MT (Metric tons)(frequency)
2001	25
2002	30
2003	32
2004	36
2005	35
2006	35
2007	37

In the above frequency table example we see that it is relative easy to understand the data. By merely looking at the table we can say that the production in the year 2004 was 36 MT. Also it is fairly simple to calculate the mean, median mode etc as well. Thus we see that representing frequency distribution in tabular has many merits.

(c) Diagrammatic representation of data: Another alternative and attractive representation of statistical data is provided by charts, diagrams and pictures. The various forms of diagrammatic representations are bar chart, histogram, pie chart, etc.

Concepts of Trigonometric Equations

In this article we have to discuss about various trigonometric equations. Before this we have to know about trigonometric identities. In mathematics trigonometric identities means which involves trigonometric functions and they show right value for every single variable. In other way we can also say that, geometrically these are the identities which involve some functions with one or more angles. They are also triangle identities which involves angle as well as side length of triangles. These identities are very useful when we solve the trigonometric-equations. One example of these identities is integration by substitution method.

Now we come to trigonometric-equations. An equation means algebraic functions written by using sum and difference formula. Similarly trigonometric-equations also contains algebraic equations but with trigonometric functions of the variables. If they contain only such functions and variables, then in solution we have to find an unknown constant which is an argument to a trigonometric function. Basically there are fundamental forms of trigonometric-equations, they are

1. Sin X=n
2. Cos X=n
3. Tan X=n
4. Cot X=n
5. Cosec X=n
6. Sec X=n

Where (n) is a constant. Now the equation sin X=n, solved only when (n) lies between the interval [-1, 1]. Suppose n comes under the interval then we need angle. After using some identities we can solve the equation. Similarly for cos X=n solved only when (n) lies between [-1, 1]. Next procedure is same. For equation tan X=n and cot X=n, (n) should be any real number. For equations cosec X=n and sec X=n, these equations can be solved by converting in other basic equations such as (cosec x=1/sin x) and (sec x=1/cos x).

Some other basic form of trigonometric-equations is (Asin x+ bcos x=0). To solve trigonometric equations using various relations between the trigonometric functions, we have to convert trigonometric-equations in a form such that value of one of the trigonometric functions of the desired variable can be determined. The roots of these trigonometric functions can be found by using inverse trigonometric functions. For solving trigonometric equations like (Asin x+bcos x=0), we have to convert this equation into basic form of equation and then using various trigonometric identities we can solve the equation. We separate this equation into real part and angle part.

Trigonometric equations solver means a process by which result can be displayed.  It can find all the solutions. The result shown first in the form of radians and then in decimal form. They show the results also in form of degrees. By using trigonometric equations solver we can solve equations easily.

Mean Median and Mode

What is the definition of mean median and mode?
Mean is a parameter of central tendency measure. The central tendency measure indicates the average value of data, where the term average is a generic term used to indicate a representative value that describes the general centre of data. Mean for raw data and for grouped data. For grouped data it is known as frequency distribution. Mean is the central value of the distribution in the sense that positive and negative deviations from the mean balance each other. It is a quantitative average.

Median is the central value of the distribution in the sense that the number of values less than the median is equal to the number of values greater than the median. So, median is a positional average. Median is the central value in the sense different from the arithmetic mean. In case of the arithmetic mean it is the numerical magnitude of the deviations that balances but, for the median it is the number of values greater than the median which balances against the number of values of less than the median.

Mode is defined as the value of the variable which occurs most frequently. Mode is for both raw data and grouped data. In raw data the most frequently occurring observation is the mode that is data with highest frequency is mode. If there is more than one data with highest frequency then each of them is a mode. Thus we have unimodal means single mode, bimodal means two mode and trimodal means three modes data sets. In grouped data mode is that value of (x) for which the frequency is maximum. If the values of (x) are grouped into the classes such that they are uniformly distributed within any class and we have a frequency distribution than we calculate maximum frequency by using formula.

What is the mean median mode in math?
Now we discuss about what does mean median mode mean in math. There are some formulas in math for mean median and mode.
For raw data mean= [∑x/n], x is refers to value of observation, n is the number of observations.
For grouped data mean= [∑fx/∑f]
Median= [(n+1)/2]
Mode=L+ [f0-f1/2f0-f1-f2]*h, L is lower limit, f0 is largest frequency, f1 is preceding frequency, f2 is next frequency and h is width.
Now, How to find the mean median and mode. Suppose we have given some problem then to find mean median and mode we can use above formula, by using formulas we can get the results.

Understanding the box and whisker plot

Box and whisker plot definition:
The basic assumption in statistics is that a set of data has a central tendency. That means the number of the data are distributed around some central value. The box of the box-whisker plot takes care of the middle half observations of the data. So we can define the box-whisker plot as a diagram that represents the tendency of data to centre on the median.

Box and whisker plot examples:
The following pictures show box and whisker plot of various data sets.

Shown above is a box-whisker plot of weight in pounds of players.

Shown below is a box- whisker plot of the length of fish in a particular lake:

From the plot we can see that the median length is 12 cm. The smallest fish has a length of 5 cm, the longest fish is 20 cm long. Most of the fishes lie between 8.5 to 14 cm in length, where 8.5 is the lower quartile and 14 cm is the upper quartile.

How to make a box and whisker plot:
Step 1: Arrange the data in ascending order from the lowest to highest value.

Step 2: Find the median of the data. For a data of n observations, median = ((n+1)/2)th observation if n is even and if n is odd then median = average of (n/2)th and ((n+2)/2)th observation. So if number of observations = 11 then (11+1)/2 = 12/2 = 6th observation is the median. But if number of observations is 14, then average of 14/2 = 7th and (14+2)/2 = 16/2 = 8th observation would be the median.

Step 3: The median that we found above has now divided the data into two halves. Now to further divide each of the half into two quarters, we need the 1st and the 3rd quartile values. Suppose the median is the mth observation. Then the middle value between the 1st and the mth observation would be the 1st quartile and the middle value between the mth and the last observation would be the 3rd quartile. They are denoted by Q1 and Q3.

Step 4: Now  you have these three values, Q1, M (= median) and Q3. We also have the lowest and the highest observation, X1 and Xn. So we now have 5 points. We make a number line using suitable scale and mark these 5 points and draw a box around the Q1 to Q3. That is our box and whisker plot.

Step by step math solution

Mathematics is used throughout the world and it is an essential tool in many fields like natural science, engineering, medicine, and the social sciences. "Mathematics" is the Greek word its gives the meaning of learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.

Step by Step Math Solutions - Method to Solve:

There are many formulae abounded for solving math problem step by step One of the most important formula is P.E.M.D.A.S. It is nothing but the operations done in the problems solving in math. They are

• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Step 1: First perform the operations inside a parenthesis.
Step 2: Now solve the exponents
Step 3: Then multiplication and division is done from left to right
Step 4: Then addition and subtraction is done from left to right

Step by Step Math Solutions - Example Problems:

Step by step math - Problem 1:

Solve the equation step by step

X + 6 = 8

Solution:

It is one step equation

Subtract 6 on both sides

X + ( 6 - 6 ) = (8 - 6)

X = 2

Step by step math - Problem 2:

Solve the equation step by step

4x + 2=18

Solution:

Subtract 2 on both sides

4x + (2 - 2) = 18 - 2

4x = 16

Divide by 4 on both sides

`(4x)/4` = `16/4`

X = 4

Step by step math - Problem 3:

Find the value of x step by step

6x - 8 = 4 x -10

Solution:

Subtract 4x  from both sides of the equation

2x - 8 = - 6

Add 6 to both sides of the equation

2x = 2

Divide both sides by 2:

x = 1

The answer is x =  1

Step by Step Math Solutions - Probability Problems:

Step by step math - Problem 1:

A fair coin is tossed two times. What is the probability of getting at least one heads.

Solution:

Let A = be the event of getting at least two heads

Let S = Sample Space which refers to the total number of probable outcomes.

S = (HH, HT, TH, TT) =4

A = (HT, TH) =2

P (A) = `2/4` = `1/2`

The probability is `1/2` .

Step by step math - Problem 2:

Consider a die is rolled; calculate the probability of in receipt of odd numbers?

Solution:

There are six different outcomes 1,3,5,7,9,10,12,14,16,17,18

n (A) = 6, total number of odd numbers occur 6.

n (S) = 11, total number of outcomes is 11.

Probability of the event A happen = P (A) = n (A) / n(S) = `5 / 11` = 0.45.

Probability of the event A does not happen = P (A') = 1 – P (A) = 1 - 0.45 = 0.55.

Circle Division

A circle is a closed curve. The points on the circle are equidistant from the center of the circle. A circle creates 2 regions, the interior of the circle is one region and the exterior is another region.

A circle can be divided into equal or unequal parts. The division can be accomplished by straight lines or curved lines. With every nth line the circle gets divided into n+1 parts, meaning with one line circle can be divided into 2 parts, with nth line circle can be divided into n+1 parts assuming that the lines do not intersect.

Circle Division Term in Maths:

A circle division can be accomplished by using a straight line or a curved line. As we are not considering the area calculation for this discussion, let us restrict ourselves to straight lines for the purposes of this study.

The straight line that joins a point on the circumference of the circle to another point on the circumference of the same circle is called a chord. A chord divides the circle into 2 parts.

Consider several chords on the circle and suppose some or all of the chords intersect. Then it is found that the number of area regions that the chords cut the circle into is always more than the number of chords passing through the circle.

Here is an example

The above is popularly known as the cake cutting example.

In the first picture, one chord divides the circle into 2 parts.in the second picture, two chords divide the circle into 4 parts, in the third three chords divide the circle into 3 parts and the forth diagram 4 chords divide the circle into 11 parts.

Here you see that the first cut creates 1 new region (1+1)

The second cut creates 2 new regions (2+2)

The third cut creates 3 new regions (4+3) and the 4th cut creates 4 new regions (7+4)

This is mathematically represented as

F (n) = n + f (n-1)

Conclusion for Circle Division:

Circle division is an important study in geometry that gives several clues for practical applications. There are several theorems and postulates that delve deeper into circle division and provide new insights into this area of study.