Trigonometry Table

Basic concepts of trigonometry , trigonometric functions , trigonometric table

The distances or heights can be found by using some mathematical techniques, which come under a branch of mathematics called ‘trigonometry’. The word ‘trigonometry’ is derived from the Greek words
‘tri’ : meaning three ,
‘gon’ : meaning sides and
‘metron’ : meaning measure

In fact, trigonometry is the study of relationships between the sides and angles of a triangle.
some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle.
The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

Application of trigonometry in real world
• In ancient time it was used for astronomy in finding distance of stars
• Finding radius of earth
• Finding height of hills, buildings, trees etc.
• Navigation – Airplane, Ships etc.
• Defense
• in architecture

Trigonometry has many applications in the modern world .without it many modern industries and sciences would simply not exist.
Just imagine , how humans could measure the height of Mount Everest!
It would be practically impossible for a person to hold a measuring tape and climb the mountain to measure the height.
Then how did they do it?
Trigonometry comes in the picture for it

Six Trigonometric Functions
There are three basic trigonometric functions which are :-
1. Sine Function (Sin)
2. Cosine Function (Cos)
3. Tangent Function (Tan)

The other three trigonometric function are reciprocal of the above written trigonometric fuction
1. Cosec Function( Cosec) { Reciprocal of Sine Function}
2. Secant Function (Sec) { Reciprocal of Cos Function}
3. Cotangent Function (Cot) { Reciprocal of Tangent Function}

Finding trigonometric functions in a right angled triangle

Special names of sides in a right angled triangle
Hypotenuse :- The hypotenuse of a right triangle is always the side opposite the right angle. It is the longest side in a right triangle.
Perpendicular :
It is the side opposite always to the reference angle given in the image.

Base (adjacent to angle theta(θ) :
The side containing the given base angle.
There are three basic trigonometric functions which are

Trigonometric Functions which are the reciprocal of the three trigonometric functions
The cosecant or cosec(A), is the reciprocal of sin(A)
It is defined as the ratio of the length of the hypotenuse to the length of the opposite side.

Also ,
1/sinA=cosecA

4. The secant sec(A) is the reciprocal of cos(A)
It is defined as the ratio of the length of the hypotenuse to the length of the adjacent side.

Also ,
1/cosA=secA

5. The cotangent cot(A) is the reciprocal of tan(A)
It is defined as the ratio of the length of the adjacent side to the length of the opposite side

NOTE :
The trigonometric function Sin A is used as a short form for “the sine of the angle A” which means ratio of perperdicular and hypotenuse .sin A is not the product of “sin” and ‘A’ as “sin” separated from the angle has no meaning.
Similarly, all other trigonometric fuction such as cos A ,tan A , cosec A , sec A and cot A is not the product .
If the angle remains the same then the values of the trigonometric ratios of an angle do not change/vary with the lengths of the sides of the triangle.
If any one trigonometric ratio of an angle is known , then by using Pythagoras theorem third side can be calculated using relation h^2=p^2+b^2 and all the other trigonometric ratios of the angle can be determined thereafter.
Trigonometric Tables are some standard values of trigonometric ratios of specific angles namely
0° , 30°, 45°, 60° and 90° .There are simple tricks and logic to remember so many values and by memorising only 5 values other 25 can be found easily .

Trigonometric formulas of complementary angles
Before learning trigonometric tables
lets revise the trigonometric formulas of complementary angles .
sinx=cos(90°-x)
cosx=sin(90°-x)
tanx=cot(90°-x)
cotx=tan(90°-x)
secx=cosec(90°-x)
cosecx=sec(90°-x) 

The above written relation can be used to establish the values of standard angles as

for example∶sin30°=cos(90°-30°)=cos60°

So the value of these two will be the same and likewise many trigonometric tables values keeps on repeating on two or more places.

Trigonometric Table of most commonly used angles.

Trigonometric tables are generally used in finding height and distances .Commonly for the specific angles 0°, 30°, 45°, 60°, and 90° as shown in a trigonometry table ,the values of six trigonometric ratios are used to solve the trigonometry problems. These values are derived by finding the ratio of the lengths of the right angled triangle and the angles of a right-angle triangle. In trigonometry , for calculation of ratio it extremely important to recognise the reference angle as in accordance to it only the six trigonometric ratios are calculated.

Thus, 0°, 30°, 45°, 60°, and 90° are called the standard and specific angles in trigonometry.

The trigonometric table consists of the values of trigonometric ratio of specific standard angles  including 0°, 30°, 45°, 60°, 90° in a table format. It is an extremely trivial work to calculate the values of the trigonometric ratios in these trigonometric table and design a patterns to remember them by  heart.There are simple ticks and tricks for remembrance of making it with ease. The table consists of the values of all six trigonometric ratios at these standard angle 0°, 30°, 45°, 60°, 90°.

TIPS AND TRICKS TO REMEMBER TRIGONOMETRIC TABLE : There are 30 values to be memorised of which 5 can be learnt and others can be found. Even those five values have a conceptual trick .
Now we will create the table using a pattern which will help to memorise all values in a very less time.

KEY TO REMEMBER TRIGONOMETRIC TABLE
Easiest method of finding Trigonometric Table
The steps given below should be followed to make the table anywhere ,anytime independently .
This table is also known as Sin Cos table.

STEP 1 : creating the table by writing angles and trigonometric angles
Create the rows & columns and make the space for writing the components of the table with the angles as mentioned 0°,30°,45°,60° and 90° in the top row and all the six trigonometric ratios in order in the first column . Firstly all the three main ratios followed by their reciprocals as shown in the table above and stated here : sinθ,cosθ,tanθ,cosecθ,secθ,and cotθ .

STEP 2 : Assigning the angles
Write the angles namely 0°,30°,45°,60° and 90°in ascending order and assign them values 0,1,2,3,4 according shown to the diagram given below.

By these simple logic and understanding remembering the trigonometric table becomes extremely easy to learn.

Frequently asked questions .

Question. What is trigonometry?
Answer. Trigonometry is a branch of mathematics that establish relationships between the three sides and angles of a triangle. It is expressed in the ratio form .Trigonometry defines trigonometric functions like sinθ,cosθ,tanθ,cosecθ,secθ,and cotθ specifically.

Question. What are trigonometric identities?
Answer. Trigonometric identity is an equation involving trigonometric ratios of an angle . The relationship is established using Pythagoras theorem.
Trigonometric identity if it is true for all values of the angle.
Important trigonometric identities:
(i) sin² θ+cos² θ= 1
(ii) 1+ tan² θ = sec² θ
(iii) 1 + cot² θ= cosec² θ

Question. What are different types of trigonometry ?
Answer. There are four types of trigonometry which is used Nowadays namely
a. Core trigonometry
b. Plane trigonometry
c. Spherical trigonometry and
d. Analytic trigonometry

Question. What are trigonometric tables?
Answer. The trigonometric table is a table which contains the collection of the values of some specific angles namely 0°, 30°, 45°, 60°, and 90°. The values of all trigonometric functions at the specified angles are given in this table.This table is crucial to be learnt for solving many types of trigonometric problems. The angles used can be expressed in degrees as well as radians . The commonly used practice is in degrees.

Question. What is the need of trigonometric tables?
Answer. Trigonometric tables are very useful in solving trigonometric problems involving trigonometric ratios of specific angles . The wider use of these table is in developing new technology specially designing computer programming and digitalization. Navigation, Civil Engineering , Architecture, mechanical engineering and many more fields. The Trigonometric tables are not restricted to only some specific angle , it can be extended to angles more that 90 degrees.

Question. Why trigonometry is used in measuring height and distances ?
Answer. Trigonometry has a widespread application in finding the measure of heights without actual measurement . They use the trigonometric ratio to establish the relationship between the given side , the unknown side and the angle opposite to 90 degrees , several examples can be written like finding the height of the mountain , tower , building , hills , height at which plane is flying , height at which balloon or kite is flying and many more.