Trigonometry Made Simple

Introduction to Trigonometry:
Trigonometry is a derived from a Greed word ‘tri’ (meaning three) and ‘gon’ (meaning sides and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early Astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on Trigonometric concepts.

The Trigonometric Ratios of the angle A in right triangle ABC are defined as:
Sine of angle A = (side opposite to angle A)/hypotenuse; cosec = 1/Sine
Cosine of angle A = (side adjacent to angle A)/hypotenuse; sec = 1/cosine
Tangent of angle A = (side opposite to angle A)/(side opposite to angle A); cot =1/tan
The trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the lengths of its sides

Trigonometry Problems and Answers:
Problem: Given angle A=51 degrees, Adjacent side length = x and opposite side length = 10. Find x and H, hypotenuse of the triangle
Answer: tan(A) = opposite side/adjacent side
tan(51)= 10/x
X = 10/(tan 51) = 8.1 (two significant digits)
Sin(A) = opposite side/hypotenuse
Sin(51) = 10/H
H = 10/ Sin(51) = 13 (two significant digits)
x=8.1 and H=13

Problem: If sin 3A = cos (A- 26 degrees), where 3A is an acute angle, find the value of A
Answer:  sin 3A can be written as cos (90-3A)
      So, we get            cos (90-3A) = cos(A-26degrees)
Since both 90-3A and A-26 are both acute angles,
90 – 3A = A- 26
4A = 116
A = 29 degrees

Problem:  An observer 1.5m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from his eyes is 45 degrees. Calculate the height of the chimney.
Answer: Let us draw a rough triangle ABC with the right angle at B. let us draw a line DE parallel to BC such that AB (AE+EB) will be the height of the chimney,  CD (equal to BE) the observer and Angle ADE, the angle of elevation.  Here, ADE is the right triangle, right-angled at E
We have, AB = AE +EB = AE +1.5 DE = CB = 28.5 (distance from the chimney)
Let us use the tangent of the angle of elevation
tan(45degrees) = AE/DE
1 = AE/28.5
AE= 28.5
Height of the chimney = AE +EB =28.5 +1.5 = 30 m

Parallelogram Definition

Let us learn about the Parallelogram Definition general, the diagonals of a parallelogram are having different lengths. The two diagonals in the figure which intersects at a particular point and lie in the interior part of parallelogram. When two pairs of the sides are opposite and they are parallel to each other.Then it is called as parallelogram .Now let us see about the parallelograms sides introduction.In parallelograms introduction, we can draw a pair of parallel lines. Draw another pair of parallel lines intersecting the former.Thus the parallelogram can be formed.Thus we can say that the pair of opposite sides of parallelogram is of equal length. Similarly we can also learn about other topics such as types of lines.Hope you like the above example of Parallelogram Definition.Please leave your comments, if you have any doubts.

Absolute Measures of Dispersion

One can define dispersion and measure it in different ways. The common dispersion definition is: “way to measure variation of data”. In general, if the dispersion is large then the variation of variable values will be huge. If the dispersion is small, then the variation is close.

Different Measures of Dispersion
The measures of dispersion can be classified into two types namely Absolute measures of dispersion and Relative measures of dispersion. Rest of the article covers these two types in detail.

Absolute Measures of Dispersion
The statistical data observations will generally be specified in some units. If the dispersion values have to be specified in the same units as that of the statistical data, then you have to choose an absolute measure of dispersion. For example, if the statistical data is represented in grams then the measure of dispersion must also be in grams.

There are various absolute measures of dispersion. Commonly used absolute measures are:
• Range
• Mean Deviation
• Variance and Standard Deviation

Here is an overview and example of one absolute measure called Range:

The Range is the simplest measure of dispersion. It is the difference between the top most value and the lowest value in the statistical set of data provided. For example, Jackson took 5 Mathematics tests in one period of time. The test marks are: 70, 60, 50, 80, and 90. What is the range of the marks? The range is calculated by subtracting the lowest value 50 from the highest value 90 and the result will be 40. Thus, the range is 40.

Relative Measures of Dispersion
If you have to compare dispersion of two or more statistical data sets which are of different units, then you have to opt for relative measures of dispersion. These dispersion measures are dimensionless and they are used to establish dispersion relation between any data sets. Common relative measures of dispersion are:
• Coefficient of Range
• Coefficient of Mean Deviation
• Coefficient of Variation
• Coefficient of Standard Deviation

Here is an overview and example of one absolute measure called Coefficient of Variation:

The coefficient of variation is the percentage of variation obtained by dividing standard deviation by mean. For example, a school has two sections for X standard. The average score of students in first section are 85 and the second section average scores are 80. The standard deviations of the two sections are 8 and 7 respectively. Which X standard section has large variation in its scores?

In this problem, the average score denotes the mean and the values of standard deviation are also specified. Now to find the variation of data by calculating coefficient of variation:

Coefficient of variance of section A is 9.4%, which is obtained by dividing standard deviation value 8 by the mean 85 and multiplying the result 0.2 by 100.

Coefficient of variance of section B is 8.75%, which is obtained by dividing 7 by 80 and multiplying the result by 100.

Comparing both the coefficients, section A is consistent when compared to section B. Hence, section B has large variation in its scores.

Harmonic Mean

Harmonic Mean

Mean is an important concept in measures of central tendency. Measures of central tendency give the measure for the center of the data. We have different types of measures of central tendency those are Mean, Median and Mode. Mean is classified into three parts those are Arithmetic Mean, Geometric Mean and Harmonic Mean. Arithmetic Mean is simply the ratio of sum of observations and the number of observations. Geometric Mean is the nth root of the product of the observations, where ‘n’ is the number of observations. What is Harmonic Mean? Harmonic Mean Definition is the reciprocal of arithmetic mean of the reciprocal of the observations. Depends on the situation we have to know which mean is correct. We have relation among these three means, that is, Arithmetic Mean = Geometric Mean = Harmonic Mean. Harmonic Mean Formula can be described as

Weighted Mean is useful in some cases where each
observation do not have an equal importance. In general arithmetic mean we are giving equal importance to each observation but this is not always the case. When all the observations are not equally important then we have to use weighted arithmetic mean. In weighted arithmetic mean we do not take sum of the observation we multiply the observations with respect to their corresponding importance. In Weighted Harmonic Mean is the reciprocal of weighted arithmetic mean of the reciprocal of observations.

Harmonic Means may not applicable in all cases if zero value present in the observations then reciprocal of zero do not exist, hence harmonic mean also do not exist. Harmonic Mean is useful when there are extreme values present in the data then it gives true picture of the average of the data. The Harmonic Mean is better average when the numbers are defined in a relation to some unit.

 For example in case of averaging speed Harmonic Mean is better measure than Arithmetic Mean. Suppose we have to find the average speed of a person travelling from place A to place B. If the person travelling with 10kmph in first hour and 15kmph in second hour then the average speed is the arithmetic mean. If the person travels first half distance with the speed 10kmph and remaining distance with the speed 15kmph then Harmonic Mean is better measure then arithmetic mean. Harmonic Mean is also called as sub- contrary mean. Harmonic Mean is useful in case of finding averages involving rates and ratios.  

Metric units and conversions

Unit conversion is converting from one unit to another of the same quantity. Unit conversion is important as it uses the metric system. Unit conversion is used to convert from one given unit to the other desired unit. Unit conversion is done for length, weight, volume, temperature, energy etc.


Unit of weight
The standard unit of mass in the metric system is gram. As per the International system of units, the S.I unit of weight is same as that of force and that is Newton but the basic unit of weight is kilogram which is usually denoted as ‘kg’. The other units for weight are kilogram (kg), milligram (mg), centigram, gram (g), decagram, hectogram, ounce and pound etc. These units are used in weight conversion that is to convert one unit from another. For example:- 1 kg = 1000g or 1 g = 1000mg.

Unit of Length – Different units of length are used in different parts of the world. As per the International system of units, the S.I unit of length is metre which is usually denoted by ‘m’. For example 10 metres is written as 10 m. There are many other units of length that are used in length conversion, some of them are millimeter (mm) , centimeter (cm), kilometer (km), decameter (dm) etc. These units can be converted to other units such as 1 mm = 1/1000 m and 1 km = 1000 m or 1 cm = 1/100 m.

Unit of Volume – There are many units which are used for volume. As per the International system of units, the S.I unit of volume is cubic metre which is usually denoted as m^3. The other units that are used to represent volume are cubic centimetre , cubic metre. Volume conversion can be done by converting one unit from another like 1 cubic metre is equal to 1000 litres or 1 millilitre is equal to 1 cubic centimetre

The unit conversions chart is very important as it helps in converting one unit from another. The chart helps in converting any quantity in the given units to the desired units. The online volume conversion chart helps in converting the difficult units.

How to convert units?
Unit conversions can be done by using the following steps: -
1. Write the terms which are given in the given unit.
2. Write the conversion factor for the given and the desired units.
3. Write it as a fraction with the given units as a denominator or in the opposite direction.
4. Cancel the ‘like’ units.
5. Multiply odd units.

Matrix- Cofactor and Adjoint

Adjoint matrix
Definition: Transpose of a cofactor matrix is called Adjoint of matrix.
 Adjoint is defined only for square matrices.
It involves two operations, which can be done in either order. We can take transpose of the given matrix and then replace the individual elements by their corresponding cofactors or the other way.
Transpose of a matrix:
Matrix obtained on interchanging the elements of rows and columns is called transpose of a matrix.

Transpose of  is  Cofactor:
The value of the determinant obtained by eliminating the row and column in which the element is present is called cofactor of the element. The cofactor is preceded with + or – sign depending upon the position of the element.
Adjoint of a 3 X 3 Matrix

Cofactor of 1 is obtained as follows:
Step 1: Identify the position of the element. [(Row, column) = (i, j)]
Step 2: remove the row and column in which 1 is present.
Step 3: Write all the other elements in the determinant form
Step 4: Evaluate the determinant append with










































For finding the Adjoint of 2 X 2 matrix, though we adopt the same technique, yet there is simple way to evaluate.

Adjoint of a 2 X 2 matrix:

Matrix Adjoint:
Adjoint of a matrix is used in evaluating the multiplicative inverse of a matrix. Multiplicative inverse is defined only non-singular matrices.

Scientific notation :word problem and expanded form

We come across large numbers in our day  to day life .we can express large number conveniently expressed using exponents.every large number can be expressed as k x 10n , where k is some natural number .however , for the sake of uniformity , we write the number in the form k x 10n  where k is terminating decimal number greater than or equal to 1 and less than 10 and n is a natural number .To  express large number into standard form is scientific notation.
Scientific notation rules
1) For the positive number , move the decimal  point to the left to bring non-zero digit to the left of the decimal point .
Example of scientific notation  :To express 41460. 5 in the standard form of decimal , we have to move decimal point 4 places on the left to get 4.14605 , so 41460.5 in the standard form of decimal is 4.14605 x 104.
2) If the given number is less  than one , then move the decimal point to the right to obtain one non –zero  digit  to the left  of the decimal point. If the decimal point is moved p places to the right ,then multiply the new number by 10^-n to express the given number in the standard form of decimal
Example of scientific notation  : To Express 0.03453 in the standard form of decimal we will have to move the decimal poit 2 places on the right to get 3.453 .so , 0.03453 in the standard form of decimal  is 3.453 x 10^-2
Scientific notation word problems
Example 1: A lake covers an area of about 3.5*10^5 square feet and its average depth of lake  is about 16 feet.   Calculate  the cubic feet of water in the lake.
Solution1) Volume  of lake =  area x depth
                                             =  3.5 *10^5 x 16 cubic feet
                                            = 56 *10^5 cubicfeet
                                           = 5.6 x 10^6 cubic feet
Expanded form : The expanded  form of a number can also be expressed  in term of powers of 10 by using
10^0 = 1 , 10^1 = 10 , 10^2 = 100
Example of expanded form :
1) 6467 = 6 x1000+ 4 x 100+ 6 x10 + 7
2) 7465267= 7 x 1000000+ 4 x100000+6x10000+5x1000+2x100+6x10+7
                               = 7x10^6+4x10^5+6x10^4+x10^3+2x10^2+6x10^1+7x10^0

Solving Complex Fractions

Complex fractions are those fractions which have a fraction in numerator or denominator or in both. For example, 1/ (¾), ½ / 3, (¾ )/ (½ ) are some of the complex fractions

Let us now learn to simplify complex fractions. While simplifying complex fraction, let us consider the following example for better understanding,

Complex Fractions

Simplify the complex fraction 4a²b/(8/ab)
Here the numerator is 4a²b and
        the denominator is 8/ab
It can be written as 4a²b  ÷ (8/ab)
to simplify, we need to flip the fraction in the denominator and multiply it with the terms in the numerator as follows,
             4a²b × ab/8 (division of fractions)
Now, we can simplify the terms
           4/8 × a²b × ab
which gives us ½ a³b²


Complex fraction solver
In solving complex fractions, we can use one more method, which is the LCD method. Let us learn how to solve  complex fractions with some example problems

1.Simplify (4/5)/(2/15)
Solution: Numerator = 4/5
Denominator = 2/15
  LCD of 5 and 15 (denominators of the two fractions) is 15
Multiply LCD with each of the fractions of the numerator and the denominator
4/5 x 15 = 4 x 3 = 12
2/15 x 15 = 2 x 1 = 2
The simplified fraction, 12/2 = 6

2.Simplify (1/a + 1/b) ÷ (1/a – 1/b)
Solution: (1/a + 1/b) = (b + a)/ab
(1/a – 1/b) = (b – a)/ab
LCD in this case would be ‘ab’
(b+a)/ab x ab = (b+a)
(b-a)/ab x ab = (b-a)
the simplified fraction is (b+a)/(b-a)

3.Simplify (x²/4)/(y/x)
Solution: x²/4 .  x/y  
= x³/4y
4.How to simplify complex fraction given below
               (a+b)/(x-y) / (a² - b²)/(x² - y²)
Solution: We re-write the given complex fraction,
          (a+b)/ (x-y)  ×  (x² - y²)/(a² - b²)
we have, (a² - b²) = (a+b)(a-b)
        (a+b)/ (x-y) × (x+y)(x-y)/(a+b)(a-b)
on simplification, we get   (x+y)/(a-b)
5.simplify the complex fraction given,
4 ¾ /3 ½
Solution: Numerator = 4 ¾ = 19/4
               Denominator = 3 ½ = 7/2
LCD of 4 and 2 is 4
19/4 x 4 = 19
7/2 x 4 = 14
the simplified fraction is 19/14