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Trigonometry Table

Introduction

Trigonometry Table deals with various angles and their corresponding trigonometric ratios. The trigonometric values of various standard angles are useful to have in-hand as they are used heavily in many real-life applications in geometrical, navigational (e.g. in ships and lighthouses), and various other areas. Some standard angles have been universally used which include 0°, 30°, 45°, 60°, and 90°. These have heavy applications and are commonly used which make knowing their value even more important. 

Lighthouse and ship

Hence, a Trigonometric Table has been carefully formulated to tabulate the trigonometric values of the various trigonometric ratios of standard angles. This table is introduced in trigonometry class 10 itself.


What is a Trigonometric Table?

The Trigonometric Table is simply a collection of trigonometric values of various standard angles including 0°, 30°, 45°, 60°, 90°, sometimes with other angles like 180°, 270°, and 360° included, in a tabular format. Because of patterns existing within trigonometric ratios and even between angles, it is easy to both predict the values of the table and use the table as a reference to calculate trigonometric values for various other angles. 
The trigonometric functions are namely sine function, cosine function, tan function, cot function, sec function, and cosec function. 

The Trigonometric Ratios Table is as follows:

 𝚹 (°)

 0° 

 30°

 45°

 60°

 90°

 180°

 270°

 360°

 𝚹(rad)

 0     

 π/6

 π/4

 π/3

 π/2

 π

 3π/2

 2π

 Sine

 0   

 1/2

 1/√2

 √3/2

 1

 0

 -1

 0

 Cosine

 1

 √3/2

 1/√2

 1/2

 0

 -1

 0

 1

 Tan

 0

 1/√3

 1

 √3

 ∞

 0

 ∞

 0

 Cot

 ∞

 √3

 1

 1/√3

 0

 ∞

 0

 ∞

 Cosec

 ∞

 2

 √2

 2/√3

 1

 ∞

 -1

 ∞

 Sec

 0

 2/√3

 √2

 2

 ∞

 -1

 ∞

 -1

A few things to be noted in the Trigonometry Table are,

  • There are a couple of pairs of complementary angles present like 30° and 60°. Their values can be computed using complementary formulas for the various trigonometric ratios. 
  • In some places, the value is said to be ∞. The reason for this is that while computing the values, a 0 appears in their denominator so the value cannot be defined and is said to be equivalent to infinity.
  • There is a change in the sign of values in various places under 90°, 180°, etc. This is due to the change in sign of the values of the various ratios in the different quadrants. 

Unit circle quadrants



How to Learn Trigonometric Table?

The Trigonometry table might seem intimidating at first, but it can be easily generated by only the values of sine for the 8 standard angles. 
Before generating the table, certain formulas must be followed and hence should be at the back of your head.

tan x = sin x/cos x
1/sin x = cosec x
1/cos x = sec x
1/tan x = cot x

Step 1
Write the angles 0°, 30°, 45°, 60°, 90° in ascending order and assign them values 0, 1, 2, 3, 4 according to the order. 
So,
0° ⟶ 0               30° ⟶ 1         45° ⟶ 2          60° ⟶ 3          90° ⟶ 4

Then divide the values by 4 and square root the entire value. 
0° ⟶  √0/2          30° ⟶ 1 /2        45° ⟶ 1/ √2          60° ⟶  √3/2          90° ⟶  √(4/4) 

This gives the values of sine for these 5 angles. 
Now for the remaining three use 
sin (180° − x) = sin x     
sin (180° + x) = -sin x
sin (360° − θ) = -sin x


Implying,
sin (180° − 0) = sin 0     
sin (180° + 90) = -sin 90
sin (360° − 0) = -sin 0

Sine 0    1/2 1/√2 √3/2 1    0    -1    0   

Step 2
sin (90° – x) = cos x 

Use this formula to compute values for cos x.
For eg: cos 45° = sin (90° – 45°)
            cos 45° = sin 45°

Similarly, 
    cos 30° = sin (90°-30°)
    cos 30° = sin 60°
Using this, you can easily find out the value of cos x row.

Cosine 1 √3/2 1/√2 1/2 10 -1    0    1   


Step 3
tan x = sin x/cos x
Hence, the tan row can be generated.

Tan 0 1/√3 1 √3 0 0

Step 4
cot x = 1/tan x
Use the relation to generate the cotan row

Cot √3 1 1/√3 0 0

Step 5
cosec x = 1/sin x 

Cosec 2 √2 2/√3 1 -1

Step 6
sec x = 1/cos x 

Sec 0 2/√3 √2 2 -1 -1

 

Applications of Trigonometry in Real Life

Why are we learning the trigonometric table? Why is it important for us to familiarize ourselves with trigonometry? It is because trigonometry holds a very important role in various aspects of our lives. And keeping a trigonometric ratios table ready to use can come very handy.  Some of these include

Astronomy
It is used to find distances between planets, sizes of celestial bodies and even removing parallax effects. It has had heavy applications in finding the distances between nearby stars and planets from the earth.

astronomy and trigo


Navigation
Trigonometry has been used for setting the various directions which include the primary 4 and the combinations of the four. It helps in the usage of compasses and even to approximate locations. It is most commonly used in ships to calculate distances from shores and lighthouses.

navigation and trigo


Criminology
Trigonometry is also used to approximate projectiles and trajectories. It is also used to calculate the angle at which bullets could have been fired and what could possibly have caused collisions in accidents. 


Measurements, Distance and Heights 
Trigonometry finds its application in calculating heights of buildings, large measurements of mountains using the angle of elevation and angle of depression along with the values of their trigonometric ratios. It is even used by marine biologists to calculate depths under the sea for observing different phenomena.

Computer Graphics and Video Games
The concept behind producing computer graphics for video gaming lies heavily on trigonometry. This includes the motion of characters, objects, and flow of the games. One of the most famous applications of trigonometric ratios, trigonometric values, and the various graphs of the sine, cos, tan, etc. is Super Mario. The motion of Mario jumping over obstacles is all thanks to the projectile motion and the trigonometric concepts behind it.

gaming and trigo

Science
The various trigonometric functions like sine and cos are periodic functions which are the basis of many kinds of waves like sound, light waves. And knowing the properties of sine and cos curves help us describe these waves as well. It can also be used to describe various parabolic motions and find out the various properties of parabolic trajectories.

Oceanography and Naval & Aviation industries
It is used to calculate the various heights of tides. It is even used by aviators.

tides height and trigo

Engineering
One of the main examples of the real-life application of trigonometry is in various branches of engineering. This includes civil engineering and construction, mechanical and even a little bit in electrical. 


Summary 

The Trigonometry Table is a very important application of Trigonometry. It is a collection of all the trigonometric values of the standard angles 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°. One does not need to mug up the values of the trigonometric ratios table, rather it provides a stepwise structure to find the values of all 6 trig functions.

Trigonometry has applications in various fields such as science, criminology, computer graphics, navigation, oceanology, etc. Hence, it is important for us to have such a collection of trigonometric values handy.
 


Frequently Asked Questions (FAQs)

sin 0, cos 0, tan 0, cot 0, sec 0, cosec 0?

Sine 0 

 0

Cos 0 

 1

Tan 0 

 0

Cot 0

 ∞

Cosec 0

 ∞

Sec 0

 0

sin pi/6, cos pi/6, tan pi/6, cot pi/6, sec pi/6, cosec pi/6,?

Sine pi/6

 1/2

Cos pi/6

 √3/2

Tan pi/6

 1/√3

Cot pi/6

 √3

Cosec pi/6

 2

Sec pi/6

 2/√3

sin pi/4, cos pi/4, tan pi/4, cot pi/4, sec pi/4, cosec pi/4?

Sine pi/4

 1/√2

Cos pi/4  

 1/√2

Tan pi/4

 1

Cot pi/4

 1

Cosec pi/4

 √2

Sec pi/4

 √2

sin pi/3, cos pi/3, tan pi/3, cot pi/3, sec pi/3, cosec pi/3, ?

Sine pi/3

 √3/2

Cos pi/3  

 1/2

Tan pi/3

 √3

Cot pi/3

 1/√3

Cosec pi/3

 2/√3

Sec pi/3

 2

sin pi/2, cos pi/2, tan pi/2, cot pi/2, sec pi/2, cosec pi/2, ?

Sine pi/2

 1

Cos pi/2

 0

Tan pi/2

 ∞

Cot pi/2

 0

Cosec pi/2 

 1

Sec pi/2

 ∞

sin cos table?

A sin cos table is another name for the Trigonometry Table which consists of all the trigonometric values of ratios of standard angles including sine function and cosine function.

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