Sine Cosine Cotangent

Sine:

          A trigonometry functions of an angle. The sine of an angle theta shortened as sin theta In a right angled triangle is the ratio of the side opposite angle to the hypotenuse. This definition applies only of angles between 0 to 90 (0 and `pi/2 ` radians).

Sin `theta` = Opposite / hypotenuse  = `(BC)/(AC)`

Looking out for more help on Sine and Cosine Identities in algebra by visiting listed websites.

Cosine:

A trigonometry function of an angle.The cosine of an angle `theta` abbreviated as cos `theta` In a right angled triangle is the ratio of the side adjacent to the hypotenuse.

Cos `theta` = Adjacent / hypotenuse = `(BC)/(AC)`

cotangent:

A trigonometry function of an angle.The cotangent of an angle `theta` ( cot  `theta` ) in a right angled triangle is the ratio of the side adjacent to it to the opposite side.
Cot `theta` = adjacent / opposite = `(BC)/(AB)`

Example problem for sin

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

we want to find the length of side c, the hypotenuse.

Here, we know that side a has a length of 8 and side b has a length of 6.
To find the length of side c, we can use the Pythagorean Theorem which says that c²=a²+b², or
Substitute in that a=8 and b=6, we find that:

c = √ (( 8²) + (6²))

  = √ (64 + 36)

  = √ 100

c = 10 m

So the value of x is found as x = 10 m
Now we have to find the value of `theta` . we can use the sin function to find the value of `theta`
Sin `theta`     = Opposite / hypotenuse  
          = 6/10
          = 0.6
Sin `theta`  = 0.6
  `theta` = sin^-1 (0.6)
`theta`  = 37<sup>o


Algebra is widely used in day to day activities watch out for my forthcoming posts on Statistics Hypothesis Testing and Integers Number Line. I am sure they will be helpful.</sup>

Example problem for cosine, cotangent:

Find the cosine and cotangent function of the given right angled triangle.

Solution:
Here we have to find the cosine and cotangent of the given right angled triangle
Cosine`theta` = Adjacent / hypotenuse
               =  4 /  5

= 0.8

cos `theta`  = 0.8
`theta` = cos^-1 (0.8)

      = 36^o
Cotangent `theta` = adjacent / opposite =  4/3

= 1.33

cot`theta`  = 1.33
`theta` = cot^-1 (1.33)

Steps in Data Analysis

Analysis of a data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has been the method of multiple facets and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains. And now we see about the steps in data analysis. (Source -Wikipedia).

I like to share this Define Quantitative Data with you all through my article. 

Two tyoes of data analysis are shown below:

• Qualitative Analysis types :

                         The reports are analyzed with the help of given data and that are supported by the qualitative Analysis    .

•  Quantitative Analysis:

                      The reports are  believed in the format of calculating types are called as the  quantitative Analysis . And it functions in the order of  normal information.
Process in data analysis types:
The some of the process in data analysis types are :

• Steps in Data cleaning:

       Data cleaning types are the is a main types for the interval of in which the data are experienced, and False  data are if important, preferable, and realistic corrected.

• Steps in Preliminary Data analysis :

       Steps in  preliminary data analysis are,

• Steps  in Quality of Data: Data can be used in a   set of  types,

         Like  Histogram, Standard possibility in plotting mean,Average variation,Median

• Steps  in Quality  analysis of size in methods:

                   The assessment of homogeneity, which provide a deference of the reliability of a component tool 

• Main steps in  data analysis:

       The main difference between of  the preliminary data analysis and the main analysis are they can get the information’s side by side.

• Types in final data analysis :

     During the final stage of  period   , the requirements  of the  of the preliminary data analysis are predictable.


Algebra is widely used in day to day activities watch out for my forthcoming posts on Simplify Using Positive Exponents and Mixed Fraction to Decimal. I am sure they will be helpful.

Example in Data analysis steps:

Example 1:
Represent the following data  in statistical histograms given below
Marks                             No.of students
0 - 5                                        2
5 - 10                                      3
10 - 15                                    4
15 - 20                                    6
20 - 25                                    2
 Solution:
Step 1: To mark the class intervals along the x – axis.
              Student marks intervals can be , 0-5, 5-10, 10-15, 15-20, and 20-25.
Step 2: To mark the frequencies along the y – axis.
               Frequencies for number of students are,4,2 , 4, 6 and 2.
               Therefore, the statistical histograms learning to depict the scale in the X-Y plane :
                                    x - axis 1 cm = 5 marks.
                                   y - axis 1 cm = 1 students
Step 3:. Total frequency = 2 + 3 + 4 + 6 + 2 = 17
           And the bar graph are drawn below:

                                

Inverse Correlation Definition

Correlation is a statistical tool which measures the degree and the direction of relationship between two or more variables.Thus correlation is a statistical device which helps us in analysing the covariation of two or more variables.

Types of Correlation

Correlation is described or classified in several different ways. Three of the most important ways of classifying correlation are:
(i) Positive or Negative(Inverse)
(ii)Simple, partial and multiple.
(iii)Linear and non-linear.

Positive and Negative Correlation.
Whether correlation is positive(direct) or negative(inverse) would depend upon the direction of change of the variables. If both the variables are varying in the same direction, i.e., if as one variable is increasing the other on an average is also increasing or, if as one variable is decreasing the other on an average is also decreasing, correlation said to be positive.If, on the other hand, the variables are varying in opposite directions, i.e., as one variable is increasing the other is decreasing or vica-versa, correlation is said to be negative(Inverse).
In other words, when two variables move in the same direction they are said to be positively correlated and when two variables moves in the opposite direction they are said to be negitively correlated.
The following examples would illustrate the difference between positive and negative correlation.

1. POSITIVE CORRELATION

X     :           10        12        15        18        20               X:          80         70         60         40         30
Y     :           15        20        22        25        37               Y:          50         45         30         20         10
2.  NEGATIVE(INVERSE) CORRELATION
X     :           20        30        40        60        80               X:         100         90         60         40         30
Y     :           40        30        22        15        10               Y:           10         20         30         40         50

Inverse correlation

My forthcoming post is on Set Builder and Interval Notation and Variable Exponents will give you more understanding about Algebra.

In a inverse correlation, as the values of one of the variables increase, the values of the second variable decrease. Likewise, as the value of one of the variables decreases, the value of the other variable increases.

This is still a correlation. It is like an “inverse” correlation. The word “negative” is a label that shows the direction of the correlation.

 Here are some other examples of negative correlations:

1. Education and years in jail—people who have more years of education tend to have fewer years in jail (or phrased as people with more years in jail tend to have fewer years of education)

2. Crying and being held—among babies, those who are held more tend to cry less (or phrased as babies who are held less tend to cry more)

Common Factoring Fractions

A common factoring fraction is the important topic in algebra.  This method is similar to taking the denominator for comparing two or more fractions.  This method is mainly used for following types of problems. They are

• Comparing fractions
• Adding fractions
• Subtracting fractions

    In this topic we have to discuss about the common factoring fractions with example problems.

I like to share this Least Common Denominator Finder with you all through my article. 

Brief explanation of common factoring fractions – Comparing Fractions

Comparing fractions:

    We can compare two or more fractions; first we can take the common denominator for all fractions. For this we can take the common factors for all denominator values. The type of comparing fractions is mostly used in the following types of problem.

They are

• Ascending order
• Descending order

Example:

Arrange the following fractions in the ascending order `(1)/(2)` , `(1)/(4)` and `(1)/(8)` 

Solution:

     Here the denominator values are 2, 4and 8. They are not equal values. So we can take common factor for all denominator values. The common denominator value is 8.

Consider `(1)/(2)` x `(4)/(4)` = `(4)/(8)`

Consider `(1)/(4)` x `(2)/(2)` = `(2)/(8)`

Consider `(1)/(8)`

     Now the denominator is same. So we can arrange the fractions in the ascending order in the following manner,`(1)/(8)`,`(2)/(8)`,`(4)/(8)`

Brief explanation of common factoring fractions – Adding and Subtracting Fractions

Adding fractions:

    We can add the two or more fractions; first we can take the common denominator for all fractions. For this we can take the common factors for all denominator values. Then we can add the numerator values.

Subtracting fractions:

    We can subtract the two or more fractions; first we can take the common denominator for all fractions. For this we can take the common factors for all denominator values. Then we can subtract the numerator values.

My forthcoming post is on Scatter Plot Graphs and Variance Math will give you more understanding about Algebra.

Example:

Simplify the fraction 1/3 + 1/6 -1/12.

Solution:

     Here the denominator values are 3, 6 and 12. They are not equal. So we can take common factor for all denominator values. That is 12.

Consider `(1)/(3)` x `(4)/(4)` =`(4)/(12)`

Consider `(1)/(6)` x `(2)/(2)` =`(2)/(12)`

Consider `(1)/(12)`

     Now the denominator is same. So we can simplify the numerator values in the following manner,

 `((4+2-1))/(12)`= `(5)/(12)`

     These are the important types of common factoring fractions.

Least Common Multiples by Prime Factorization

The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. The short term for Least common multiples is LCM. (Source Wikipedia)

Examples for finding Least common multiples by prime factorization:

Example 1:

Find the least common multiples for the numbers 234 and 534 by prime factorization.

Solution:

Prime factorization for the given numbers are

234   :  2 3 3 13

534   :  2 3        89

----------------------------

LCM:  2 3 3 13 89

Least common multiples by prime factorization is 2 * 3 * 3 * 13 * 89 = 20826

Example 2:

Find the least common multiples for the numbers 584 and 564 by prime factorization.

Solution:

584    : 2 2 2 73

564    : 2 2 3 47

------------------------

LCM : 2 2 2 3 73 47

Least common multiples by prime factorization is 2 * 2 * 2 * 3 * 73 * 47 = 82344

Example 3:

Find the least common multiples for the numbers 124 and 164 by prime factorization.

Solution:

124    : 2 2 31

164    : 2 2 41

---------------------

LCM : 2 2 31 41

Least common multiples by prime factorization is 2 * 2 * 31 * 41 = 5084

Example 4:

Find the least common multiples for the numbers 48 and 58 by prime factorization.

Solution:

48      : 2 2 2 2 3

58      : 2            29

------------------------------

LCM : 2 2 2 2 3 29

Least common multiples by prime factorization is 2 * 2 * 2 * 2 * 3 * 29 = 1392

Example 5:

Find the least common multiples for the numbers 45 and 64 by prime factorization.

Solution:

45      : 5 3 3

64      :          2 2 2 2 2 2

-------------------------------------

LCM : 5 3 3 2 2 2 2 2 2

Least common multiples by prime factorization is 5 * 3 * 3 * 2 * 2 * 2 * 2 * 2 * 2= 2880

Practice problems for finding Least common multiples by prime factorization:

Problem 1:

Find the least common multiples for the numbers 56 and 456 by prime factorization.

Least common multiples by prime factorization is 3192

Problem 2:

Find the least common multiples for the numbers 64 and 65 by prime factorization.

Least common multiples by prime factorization is 4160

Problem 3:

Find the least common multiples for the numbers 560 and 4560 by prime factorization.

Least common multiples by prime factorization is 31920

Between, if you have problem on these topics Series Solutions  please browse expert math related websites for more help on Multiple Linear Regression Analysis.

Problem 4:

Find the least common multiples for the numbers 356 and 78 by prime factorization.

Least common multiples by prime factorization is 13884

Problem 5:

Find the least common multiples for the numbers 4562 and 5697 by prime factorization.

Least common multiples by prime factorization is 25989714

Checking Prime Factorization

Prime number is a number that is divisible by 1 and itself. Prime factorization is nothing but the factorization of a prime numbers. To find the prime factorization of a number the number it factored until all its factors are prime numbers.In this topic we are going to see problems in prime factorization and checking the factors are prime or not. Some example problems are solved below.

Prime Factorization and checking – Example problems:

Find the prime factors of 24 and check?
24 ÷ 2 = 12
12 is not a prime number, so factor again,
12 ÷ 2 = 6
6 is not a prime number, so factor again,
6 ÷ 2 = 3
3 is a prime number,
24 = 2 × 2 × 2 × 3
Checking whether 2,2,2,3 are the prime factors of 24
2 × 2 × 2 × 3 = 24
Therefore 2, 2, 2, 3 are the prime factors of 24
Example 2:
Find the prime factorization of 124 and check ?
124 ÷ 2 = 62
62 is not a prime number, so factor again:
62  ÷ 2 = 31 
31 is a prime number.
124 = 2 x 2 x 31
Checking
2 * 2 * 31 = 124
Therefore 2, 2, 31 are the prime factors of 124. 

prime factorization tree.
                      124
                      /    \
                 2         62
                           /    \
                         2    31

Some other examples of prime factorization:

My forthcoming post is on SAS Similarity and Define Box and Whisker Plot will give you more understanding about Algebra.

Example 3:
Find the Prime factor of  81 and check?
Solution:
81 ÷ 9 = 9
9 = 3 * 3
81 = 3 x 3 x 3
Checking
3*3*3 = 81
Therefore the prime factors are 3,3,3
Example 4:
Find the prime factor of 13 and check?
Solution:
39 = 3 x 13
So the prime factor of 19 is 3 and 13.
Checking
3 * 13 = 39
Therefore the prime factors of 31 are 3 * 13.

Precalculus Problem Solution

Calculus is one of the learning about the rates of change and measurement of changing quantities in which it is using symbolic notations in the calculations. In precalculus it identifies the values of the function and it does not involved in the problems of calculus but it explores topics that will be applied in calculus. So that precalculus one of the main division in the calculus

Calculus two types, they are.          

• Differential calculus
• Integral calculus       

      Precalculus functions that are given is, Domain and Range Composition, and Difference of Quotients

Differential calculus:

        Differentiations are been used to determine the rates of change of a function and the larger and smaller values of a function.

        It is a branch of Calculus that deals with derivatives and their applications.

Integral Calculus:

         Integration is the branch of the calculus in which it deals with integrals and its application,  in determining areas, equations of curves, or volumes.                                                                                                                

                          

Some precalculus operations:

• The slope of line passing through points of` (x_1, y_1) ` and` (x_2, y_2)` which is given by,

                        m = `(y2 - y1) / (x2 - x1)`

• slope m has the equations. in which passes through the line through the point (x₁, y₁).

                    y - y₁ = m(x - x₁)

• y-intercept the b has the equation of line y = mx + b.It is the line passes through the slope
• Two lines of the equation with slopes m₁ and m₂.

            If the line of the slope is parallel m₁ = m₂ and

            If the line of the slope is perpendicular m₁m₂ = -1.
                                                      

Precalculus problem solution - Example problems:

Precalculus problem solution - problem 1:

Solve the equation with the  curves y = mx, where, m is arbitrary constant.

Solution:

      We have the equation of the curve

               y = mx ... (1)

     Differentiating either side of equation (1) with respect to x, we get

      `dy / dx` = m

     Substitute the value of m in equation (1) we get

               y = ` (dy / dx)` * x

     or 

            x `( dy / dx ) ` - y = 0

    Hence this is the required differential equation.         
Precalculus problem solution - problem 2:
Find the derivatives for,   (i) Y = `4 / x^3`   (ii) Y = 6`sqrt (x^3)`
   Solution :
                  (i). Y = `4 /x^3 `
                        Y = 4 X ^-3
                     ` dy / dx` = (4-3)x^(-3-1)
    Answer     = `(1 / x^(-4))`
               (ii) Y = ` 6sqrt(x^3)`
                      Y =  `6 x^(1 /3)`
             `dy / dx` = `6 (1/3)` `x ^-(2 / 3)`
           `dy / dx ` = `(6/3)x^-(2/3)`
     Answer = `(6/3)x^-(2/3)`