## Trigonometric Formula, Ratio, and Angle Table

In Mathematics, these trigonometric tables are useful in a wide number of areas. Before we were introduced to pocket calculators, we had trigonometric tables with us. There are certain values in each trigonometric table that will help you to solve your sums and equations very easily. The trigonometric Table comprises sin, cos, tan, cosec, and sec values at different theta and here, theta is the value of the degree of angle.

If we talk about what this trigonometry is then, then trigonometry is the branch of mathematics that involves the study of relationships including that of the length of a triangle and its angles. Generally, trigonometry is associated with a right-angled triangle, a triangle in which one believes it lays at 90 degrees. It not only has used in solving mathematical problems, but it also has use in the field of navigation as well as other science and engineering fields.

The trigonometry Table comprises the values of trigonometric ratios such as sine, Cosine, tangent, cotangent, cosecant, and secant from 0 to 360°. By applying values from 0 to 360°in these trigonometric ratios, we get the following values listed in the trigonometry value table:

Here this table provides you with all the trigonometric values. This table is the base for you to solve a number of sums and equations with the trigonometric formula. It is often observed that many formulas are derived from these trigonometric values. This trigonometric table is basically the collection of all the trigonometric values but at different angles. 0° 30° 45° 60° and 90° are considered as the standard angles of trigonometric values.

### More about Trigonometric Table

We can see in the trigonometry value table the trigonometric ratios relate to each other, the variations occur in taking the different trigonometric ratios of sin, cosine, secant, cosecant, tangent, and cotangent.

On-demand, the trigonometric functions are generated by calculators and modern computers also but by using special libraries of these mathematical codes. These trigonometric tables are going to make your work much easier. You just need to learn this table and most of the sums, you will be able to solve them easily.

The trigonometric table is made up of the following trigonometric ratios that are interrelated to each other – sine, cosine, tangent, cosecant, secant, cotangent. These ratios, in short, can be written as sin, cos, tan, cosec, sec, and cot.

\[sin = \frac{perpendicular} {hypotenuse}\]

\[ cos = \frac{base} {hypotenuse}\]

\[tan = \frac{perpendicular}{base}\]

\[cosec = \frac{hypotenuse}{perpendicular}\]

The applications of Trigonometry in other associated scientific and mathematical fields is enormous with the distances on earth as well as in space being estimated in the ancient world with careful use and application of Trigonometry.

The calculations can easily be figured out by memorizing a table of functions most commonly known as the Trigonometric Table. This finds use in several areas. Some of them include navigation science, geography, engineering, geometry etc. The trigonometric table was the reason for most digital development to take place at this rate today as the first mechanical computing devices found application through careful use of trigonometry.

The Trigonometric ratios table gives us the values of standard trigonometric angles such as 0°, 30°, 45°, 60°, and 90°. These values hold increased precedence as compared to others as the most important problems employ these ratios. It is therefore very important to know and remember the ratios of these standard angles.

Memorizing the trigonometry table will be useful as it finds many applications, and there are many methods to remember the table. Knowing the Trigonometric Formulae automatically will lead to figuring out the table and the values. The Trigonometry ratio table is dependent upon the trigonometry formulas in the same way all the functions of trigonometry are interlinked with each other.

### Trigonometric Functions

Now if we talk about what these trigonometric functions are, then these trigonometric functions are the functions of a right-angled triangle. These trigonometric functions are basically six in number and they're:

Sin function

Cos function

Tan function

Sec function

Cosec function

Cot function

These functions are basically the ones that are going to help you to solve the problems and get yourself clear with a number of concepts.

### Trigonometric Functions of Complementary Angles

Before attempting to begin, it is better to try and remember these values, and know the following trigonometric formulae. Because if you are well aware of the formulas beforehand, then only it will be easy for you to attempt the questions. Every function is related to its corresponding function. Just as in the formulas below, you will observe that the sin function is related to the cos function and vice versa. Further, the is function is related to the tan function and vice versa. The other one that is sec function is also related to the cot function and vice versa.

sin x = cos (90°− x)

cos x = sin (90°− x)

tan x = cot (90°− x)

cot x = tan (90°− x)

sec x = cot (90°− x)

cot x = sec (90°− x)

1/sin x = 1/cos x = sin x

1/cos x = sec x

1/sec x = cos x

1/tan x = cot x

1/cot x = tan x

### Steps to create Trigonometry Table:

Step 1: Draw a tabular column with the required angles such as 0, 30, 45, 60, 90, in the top row and all 6 trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent in the first column.

Step 2: Find the sine value of the required angle.

To determine the value of sin we divide all the values sequentially from 0, 1, 2, 3, and 4 by 4 and then take the square root. For example, to find the value of 0°, we will write \[\sqrt{(\frac{{0}}{4})}\], i.e., 0 or to find the value of 30°, we will write \[\sqrt{(\frac{{1}}{4})}\], i.e.,\[\frac{1}{2}\]. So, the corresponding values from 0 to 360° are

Step 3: Find the cosine value of the required angle.

The cos values are tabularly opposite to that of sin angles.

This means that whichever value of sin (0 - x) degree is the same as the value of cos (90 - x) degree. To find the value of cos, divide by 4 in the opposite order of sin, i.e., sequentially from 4, 3, 2, 1, and 0 by 4 and take the square root.

For example, to find the value of 0°, we will write √(4/4), i.e., 1 or to find the value of 30°, we will write \[\sqrt{(\frac{{3}}{2})}\], i.e., \[{(\frac{\sqrt{3}}{2})}\]. So, the corresponding values from 0 to 360° are:

Step 4: Find the tangent value of the required angle.

The tangent is equal to sine divided by cosine. tan x = sin x/cos x.

For finding the value of tan 30, we divide sin 30 by cos 30 and get the required value, i.e., \[\frac{(\frac{1}{2})}{(\frac{\sqrt{3}}{2})} = \frac{1}{\sqrt{3}}\], the other corresponding values are:

Step 5: Determine the value of the cot.

The value of the cot can be determined by all inverse values of tan.

So, for every value, the cot value is 1/tan. As cot x = cos x/sin x. So the corresponding values are just the reciprocal of the tan values.

Step 6: Find the cosecant value of the required angle.

The value of cosec on any angle is the inverse of sin on that particular angle. So, the corresponding values will be the reciprocal of the values in sin x.

Step 7: Determine the value of sec.

The value of sec on any angle is the inverse of cos of that particular angle. So, the reciprocal of cos values are:

Following the above-mentioned steps, you will be able to easily create this table on your own and once you have figured out how to create this table you can easily learn it. Learning this table is as important as solving the sums using these tables because if you have learnt the table only then you will be able to use it quickly without taking any time. Solving the question will become a much easier task for you as you will know about the values beforehand and you just need to recall the value and you will get the answer.

### Trigonometry Formulas List

1. Compound Angles

cos A cos B – sin A cos B = cos (A + B)

cos A cos B + sin A cos B = cos (A – B)

sin A cos B + cos A sin B = sin (A + B)

sin A cos B – cos A sin B = sin (A – B)

sin\[^{2}\] A – sin\[^{2}\] B = sin (A + B) sin (A – B) = cos\[^{2}\] B – cos\[^{2}\] A

cos\[^{2}\] A – sin\[^{2}\] A – sin\[^{2}\] B = cos (A + B) cos (A – B) = cos\[^{2}\] B – sin\[^{2}\] A

(tan A + tan B)/(1 – tan A tan B) = tan (A + B)

(tan A – tan B)/(1 + tan A tan B) = tan (A – B)

sin2A = sin (A + A) = sinA.cosA + cosA.sinA = 2sinA.cosA

2. Sum and Differences of Sines and Cosines

sin (A+B) + sin (A-B) = 2 sin A cos B

sin (A+B) – sin (A-B) = 2 cos A sin B

cos (A+B) + cos (A-B) = 2 cos A cos B

cos (A-B) – cos (A+B) = 2 sin A sin B

3. Trigonometric Ratios of Multiples of Angles

sin 2A = 2sinA.cosA = \[\frac{2 sinA . cosA}{1}\] = \[\frac{2 sinA . cosA}{cos^{2}A+sin^{2}A}\] = \[\frac{\frac{2sinAcosA}{A}}{\frac{A}{A}+\frac{A}{A}}\] = \[\frac{2tanA}{1+tan^{2}A}\]

sin 2A = \[\frac{2tanA}{1+tan^{2}A}\] = \[\frac{\frac{2}{\text{cos cosA}}}{1+\frac{1}{A}}\] = \[\frac{2}{\text{cot cotA}}\] х \[\frac{A}{1+cot^{2}A}\] = \[\frac{2cotA}{1+cot^{2}A}\]

cos 2A = cos(A+A) = cosA.cosA - sinA.sinA = cos\[^{2}\]A - sin\[^{2}\]A

cos 2A = cos\[^{2}\]A - sin\[^{2}\]A = 1 - sin\[^{2}\]A - sin\[^{2}\]A = 1 - 2sin\[^{2}\]A

cos 2A = A - A = A - (1 - A) = 2A - 1

cos 2A = A - A = \[\frac{A-A}{1}\] = \[\frac{A-A}{A+A}\] = \[\frac{\frac{A}{A}-\frac{A}{A}}{\frac{A}{\omega_{q}^{2}A}+\frac{A}{\omega_{l}^{2}A}}\] = \[\frac{1-tan^{2}A}{1+tan^{2}A}\].

cos 2A = \[\frac{1-tan^{2}A}{1+tan^{2}A}\] = \[\frac{1-\frac{1}{A}}{1+\frac{1}{A}}\] = \[\frac{cot^{2}A - 1}{cot^{2}A + 1}\]

tan2A = tan(A + B) = \[\frac{tanA+tanA}{1-tanA.tanA}\] = \[\frac{2tanA}{1-tan^{2}A}\]

tan 2A = \[\frac{2tanA}{1-tan^{2}A}\] = \[\frac{\frac{2}{\text{cot cotA}}}{1-\frac{1}{A}}\] = \[\frac{2}{\text{cot cotA}}\] х \[\frac{A}{cot^{2}A - 1}\]

Cot 2A = cot2A = cot(A+A) = \[\frac{cotA.cotA-1}{cotA+cotA}\] = \[\frac{cot^{2}A-1}{2cotA}\]

Cot 2A = \[\frac{cot^{2}A-1}{2cotA}\] = \[\frac{\frac{1}{A}-1}{\frac{2}{\text{tan tanA}}}\] = \[\frac{1-tan^{2}A}{A}\] х \[\frac{1-tan^{2}A}{2tanA}\] =\[\frac{1-tan^{2}A}{2tanA}\]

4. Multiples and Submultiple Angles

2 sin A cos A = 2 tan A/(1 + tan\[^{2}\] A) = sin 2A

cos 2A – sin2A = 2cos2A - 1 = 1 - 2 sin2A = cos 2A

4 cos 3A – 3 cos A = cos 3A

3 sin A – 4 sin 3A = sin 3A

2 tan A/(1 - tan\[^{2}\]A) = tan 2A

(3 tan A- tan\[^{3}\] A)/(1 - tan\[^{2}\] A) = tan 3A

## FAQs on Trigonometry Table

**1. Give a brief overview of trigonometry.**

Trigonometry is a branch of Mathematics, that deals with establishing relationships between lengths of sides and angles of a triangle. It has wide applications in the field of oceanography primarily. Moreover, there are other applications in real-life areas such as music and other sound waves.

The founding of Trigonometry was very useful because a computer cannot recognize musical sounds just as we do. Although, it could be specified in the same format or way in a Mathematical form as a function of sine or cosine which can be developed into musical sounds. Three prominent features are:

Height

Base

Hypotenuse

In this part of trigonometry, you will be taught about some of the standard angles in Trigonometry, which are 0o, 30o, 45o, 60o, and 90o. It is mandatory to know these fundamental angles as they are mostly used in numerical.

**2. What are the trigonometric ratios and applications of trigonometry.**

Trigonometry ratio is defined as the relationship between the measurement of lengths and angles of the right triangle and also be used in particular areas of the circle.

Trigonometric ratios in trigonometry are derived from the three sides of a right-angled triangle, basically the hypotenuse, the base(adjacent), and the perpendicular(opposite).

These trigonometric identities and trigonometric formulas are used widely in all science-related topics such as mechanics, geometry, and many others.

A few applications of Trigonometry are

Trigonometry is used in cartography which is the creation of maps.

It has its applications in satellite systems.

It is used in the aviation industry.

The functions of trigonometry are used to describe sound and light waves.

**3. Where do you use Trigonometry in your day-to-day life?**

We may not use it directly in our everyday life, but there are a lot of solutions to solve practical issues which are to be used in various things that we have so much fun with. For instance, you listen to music every day, but we do not observe the waves which travel from sound and this is exactly how Sine or Cosine functions. This is also a method used for developing music on the computer. The biggest stress buster of our everyday life is Music. But, how many of us know about Trigonometry being used in it?

Computers do not understand human language or music like a human. So, computers Mathematically represent waves by constituent sound waves. It is a minimum thing for any Sound engineer to know at least the basics of trigonometry.

**4. What to expect from Vedantu?**

At Vedantu, our group of faculty members are working in unison for over the years to deliver the right knowledge to the students in the most innovative ways. Our main motto is to impart knowledge to the students and help them boost their exam scores in the final examination.

The whole content designed by Vedantu is for you to increase your ability to solve problems and improve your overall performance. Vedantu provides every study material that you will require to increase your knowledge about a particular theory and besides this, it increases your ability to learn a few things. Everything that you receive from Vedantu is specially designed by the subject experts and there is hardly any chance that you will face a problem while using it.

Our NCERT Solutions gives you a clear idea and understanding of all the important concepts and helps you develop a strong conceptual foundation. They provide an in-depth view of the concept. These solutions cover all possible questions and question types that can be asked in your Math exams.

**5. How are these trigonometric tables going to help me with my studies?**

These trigonometric tables are the base of your mathematics. You are going to solve a number of problems by using these tables. They form the base for you so that you can make the other calculations easily. The table should be on your tips. You should be well aware of all the values that you have when any question is asked of you. You can answer it in minutes using these tables. This is going to help you quickly with answers and reduce your time in calculations.