Trigonometry is one of the main branch of mathematics which is a very important part of it and deals with various equations which need different trigonometric formulas and identities to be solved so today here in this article we are going to discuss in detail about different formulas and identities which are being used in mathematical equations for getting perfect solutions with accurate answers.
You will be going to get all of the basic trigonometry formulas here at single place, the diversity of trigonometric problems is very vast and with help of these simple trigonometric formulas one can easily solve these kind of problems the one thing you should keep in mind that the trigonometric problems do not get solved with just one formula it may be possible that you may require two or more formulas to solve a particular problem so it is very important to know the perfect use of these equations or identifies and for getting perfect results you have to practice them deeply again and again.
The trigonometry and its formulas basically used to deal with the triangles, sometimes it has been also seen that trigonometry is referred as the study of relation in between the length of triangle and angles of triangle apart from triangle in some cases the trigonometry also deals with the circle expressions. There are number one places where we use trigonometry and its formulas like as in the technique of triangulation which is being used for the geography for measuring distance in between different landmarks or in the astronomy for measuring the distance between the stars or in the system of satellite navigation.
The trigonometric identities are basically trigonometry formulas the only difference is that they also contain trigonometric functions. These trigonometric identities are valid for all type of variables. The another most used term is trigonometric ratio which is a healthy relationship in between measurement of angles and in the length of sides of a right angles triangle. Today here we will also provide you the trigonometric formulas PDF this will help you a lot in keeping all formulas at single place.
Trigonometric Functions
So , here we are going to discuss the trigonometric function formulas as we all know that in a right angles triangle there are basically three sides which are named as hypotenuse , opposite side to the hypotenuse is named as perpendicular and the adjacent side to the hypotenuse is named as the base. The hypotenuse is the longest side of the triangle or we can also say that the side which is in front of the right angle is known as the hypotenuse. To find the elements of trigonometry there are mainly six laws and these are known as the trigonometric functions. So, these six trigonometric functions are as follows site, cosine, secant, cosecant, tangent and the last one is cotangent so, if we use the above right angle triangle reference we can extract the value of these trigonometric functions by the following formulas let us know them :
* Sin θ = Perpendicular / hypotenuse
* Cos θ = Base / hypotenuse
* Tan θ = Perpendicular / base
* Cosec θ = Hypotenuse / perpendicular
* Sec θ = Hypotenuse / base
* Cot θ = Base / Perpendicular
* Cosec θ = 1 / sin θ
* Sec θ = 1 / cos θ
* Cot θ = 1 / tan θ
* Sin θ = 1 / cosec θ
* Cos θ = 1 / sec θ
* Tan θ = 1 / cot θ
Inverse Trigonometric Functions
Now , we are going to share some inverse trigonometric functions formulas which will help you in solving different problems :
Function Inverse
* Sin arcsin
* Cos arccos
* Tan arctan
* Cosec arccosec
* Sec arcsec
* Cot arccot
So , these all were the trigonometric identities this was just for your easy review here we are also going to provide you trigonometric functions PDF which will help you a lot in keeping these details or functions for future reference.
Trigonometry Formulas List
Now , we are going to share a small list of some of the basic trigonometric formulas which are used most widely and easy to learn while the complete details of these formulas based on different terms will be given in PDF of it. So, here is the list of trigonometry formulas :
* Sin ( 90° – x ) = cos x
* Cos ( 90° – x ) = sin x
* Tan ( 90° – x ) = cot x
* Cot ( 90° – x ) = tan x
These were the trigonometric formulas based in degree which includes cofunction identities.
Trigonometric Identities
After getting information about trigonometric formulas we are going to tell you about the trigonometric identities as these are the very important part of trigonometry which includes the trigonometry formulas and functions. We use these identities for finding some functions for one or more angles within equation and are very useful on those places where trigonometric functions are needed to be simplified from there original state or form. Here we are also going to provide you trigonometric identities PDF which will keep all of the basic information at one place safely. Let us come to know about some of the trigonometric identities :
Trigonometric Identities class 10
There are not very wide range of trigonometric identities which is used by the students of class 10 there are only three basic identities which they use so, these three basic identities are :
Trigonometry Table
The trigonometric table is a very important part of trigonometry as most of the solutions are derived with help of trigonometric table values. So , as we know the importance of trigonometry table therefore we are here providing you the trigonometric table 0 to 360 degree this will give you a wide range of overview and values of different angles.
Degrees

Sin θ  Cos θ  Tan θ  Cot θ  Sec θ  Cosec θ 
0 °  0  1  0  undefined  1  undefined 
30 °  1 / 2  √ 3 / 2  √ 3 / 3  √ 3  2 √ 3 / 3  2 
45 °  √ 2 / 2  √ 2 / 2  1  1  √ 2  √ 2 
60 °  √ 3 / 2  1 / 2  √ 3  √ 3 / 3  2  2 √ 3 / 3 
90 °  1  0  undefined  0  undefined  1 
120 °  √ 3 / 2  – 1 / 2  – √ 3  – √ 3 / 3  – 2  2 √ 3 / 3 
135 °  √ 2 / 2  – √ 2 / 2  – 1  – 1  – √ 2  √ 2 
150 °  1 / 2  – √ 3 / 2  – √ 3 / 3  – √ 3  – 2 √ 3 / 3  2 
180 °  0  – 1  0  undefined  – 1  undefined 
210 °  – 1 / 2  – √ 3 / 2  √ 3 / 3  √ 3  – 2 √ 3 / 3  – 2 
225 °  – √ 2 / 2  – √ 2 / 2  1  1  – √ 2  – √ 2 
240 °  – √ 3 / 2  – 1 / 2  √ 3  √ 3 / 3  – 2  – 2 √ 3 / 3 
270 °  – 1  0  undefined  0  undefined  – 1 
300 °  – √ 3 / 2  1 / 2  – √ 3  – √ 3 / 3  2  – 2 √ 3 / 3 
315 °  – √ 2 / 2  √ 2 / 2  – 1  – 1  √ 2  – √ 2 
330 °  – 1 / 2  √ 3 / 2  – √ 3 / 3  – √ 3  2 √ 3 / 3  – 2 
360 °  0  1  0  undefined  1  undefined 
Trigonometry Triangle formulas
Till now we have learnt a lot about the trigonometry its formulas, functions, and identities now we are going to learn about the trigonometry formulas in details according to different classes and the use of these formulas in those classes so let us come to learn more about them you will also be going to get trigonometry formula sheet in this article. Let us come to learn them in detail:
Trigonometry Formulas
Most of the students fear from maths as they find it more difficult than any other subject and there is one thing which is set in their mind is that maths is a very tough subject and it is not easy to score good marks in this subject. As we know that what we set in our mind mostly the same happens so in this situation they were not be able to focus on their subject and solving problems of Maths which makes them nervous when they go for the exam or sometimes they also get too much nervous during a small class test , to score good marks in Maths it is very important to let your fear out of your mind and the only way of it is practising more and more with different problems. Trigonometry is one of the such chapter which is found more difficult by 90 % of students therefore we are here going to provide you important trigonometric formulas class wise which will help you in solving all of your trigonometric problems lets start it :
Trigonometric Formulas for Class 10
We have already provided the trigonometric table above in this article which will help you in solving most of your questions now we are here going to provide you all of the major trigonometric formulas for the students of class 10 these are as follows :
* Sin θ = side opposite to angle θ / Hypotenuse = P / H
* Cos θ = side adjacent to angle θ / Hypotenuse = B / H
* tan θ = side opposite to angle θ / side adjacent to angle θ = P / B
* Sin θ = Perpendicular / hypotenuse
* Cos θ = Base / hypotenuse
* Tan θ = Perpendicular / base
* Cosec θ = Hypotenuse / perpendicular
* Sec θ = Hypotenuse / base
* Cot θ = Base / Perpendicular
* Cosec θ = 1 / sin θ
* Sec θ = 1 / cos θ
* Cot θ = 1 / tan θ
* Tan θ = sin θ / cos θ
Note: The outcome for sin and cos will never exceed than 1 and the value of sec and cosec will always be greater than 1 or equal to 1. You have to keep this point remember for getting exact values.
A . Sin ( 90° – θ ) = cos θ
B . Cos ( 90° – θ ) = sin θ
C . Tan ( 90° – θ ) = cot θ
D . Cot ( 90° – θ ) = tan θ
E . Sec ( 90° – θ ) = cosec θ
F . Cosec ( 90° – θ ) = sec θ
G . sin² θ + cos² θ = 1
H . tan² θ + 1 = sec² θ
I . cot² θ + 1 = cosec² θ
Trigonometric Formulas for Class 11
Trigonometry is the biggest fear of students and specially in higher classes this fear gets increased. The most common problem which is generally faced by students is of remembering formulas but if you get all these formulas at single place then it really becomes easier for you to remember those formulas so here we are going to provide the trigonometric formulas for the students of class 11.
* sin ( – θ ) = – sin θ
* cos ( – θ ) = cos θ
* tan ( – θ ) = – tan θ
* cosec ( – θ ) = – cosec θ
* sec ( – θ ) = – cot θ
Basic Formulas
1 . Sin ( A + B ) = sin A . cos B + cos A . sin B
2 . Sin ( A – B ) = sin A . cos B – cos A . sin B
3 . Cos ( A + B ) = cos A . cos B – sin A . sin B
4 . Cos ( A – B ) = cos A . cos B + sin A . sin B
5 . Tan ( A + B ) = tan A + tan B / 1 – tan A . tan B
6 . Tan ( A – B ) = tan A – tan B / 1 + tan A . tan B
7 . Cos ( A + B ) . cos ( A – B ) = cos² A
8 . – Sin² B = cos² B – cos² A
9 . Sin 2 A = 2 sin A . cos A
10 . Cos 2 A = cos² A – sin² A = 1 – 2 sin² A
11 . Tan 2 A = 2 tan A / 1 – tan² A
12 . Cos³ A = 4 cos³ A – 3 cos A = 4 cos
So , these all were some of the basic trigonometric formulas which are important for the students of class 11 to solve equations and the trigonometric table has already been given above in this article.
Trigonometric Formulas for class 12
The trigonometry of class 12 has various new things like as of trigonometry definitions , inverse properties and much more along with the trigonometric formulas so , here we are going to give an overview about important definitions , properties and formulas for the students of class 12 which will help them a lot during exam so , to know more have a look below :
* Sin ( 2 x ) = 2 sin x . cos x
* Cos ( 2 x ) = cos² x – sin² x
* Tan ( 2 x ) = 2 tan x / 1 – tan² x
* Sin x / 2 = ± √ 1 – cos x / 2
* Cos x / 2 = ± √ 1 + cos x / 2
* Tan x / 2 = 1 – cos x / sin x = sin x / 1 + cos x
Trigonometric Function Graph
Next we are going to provide you the graph of trigonometric functions there are basically six functions named as sin, cosine, tangent, secant, cosecant, cotangent so these all have different functions graph which we have shown in the below image.
Now , it may seem a little bit confusing to understand these graphs let us explain you a graph of sine. You will see a point at the terminal side of an angle there you can say that the sine angle is basically a ratio for the y coordinate of the particular point which will be calculated from the distance between this point and origin. Now , we will assume that angle is getting changed but the distance of point is same with respect to the origin so the point will surely trace the circumference of this circle.
As the angle changes from 0 to the π / 2 the coordinate of y will increase and also the sine angle will get increased similarly if the angle goes differently from the π / 2 to the 0 then the y coordinate will get decreased and with this the angle of sine will also get decreased. So, by putting these methods in all of these six graphs you can understand them and come to learn how to make a particular graph or how to get the detail of values or angles.
Trig Equation calculator
The trig equation calculator is nothing it is simply a trigonometry sin cos tan calculator, there are lots of online calculator which are available on various websites to make your work more easy. If you are in lack of them then in this situation you can easily use this calculator or if you feel that you need to check your answer whether it is correct or not then also you can easily use these calculators. The online equation calculators give you exact answer for any of your trigonometric problem within seconds it is very easy to use and gives accurate result. If you also wanted to get answer for your trigonometric equation then it is so simple to do so let us tell you how :
 First of all, you have to go through any of the official online trigonometric calculators here we are going to provide you the link to one such calculator which is https://www.rapidtables.com/calc/math/trigonometrycalculator.html
 By clicking on this link you will be redirected to a new window there you will see some blank boxes, in those boxes you have to fill up the necessary details which are present in the equation which you wanted to solve.
 Now, after filling those boxes with required details you have to simply click on the option of calculating which is present just below that box.
 After clicking on calculate option wait for few seconds and you will get the answer or solution for your trigonometric equation.
So , in this way you can easily calculate the value or get solution online without doing it manually , in this calculator you also have the option of Reset just at the side of Calculate by which if you wanted to cross check your answer or want to solve any other equation then you can simply click on the icon of reset and this will automatically clear all of the previous data which has been entered by you in those boxes.
It is very simple and easy to calculate trigonometric equations with help of these online calculators but if you find any difficulty then you can easily comment on your problem. So, today in this article we give you all of the necessary information related to the trigonometry and its formulas hope you will like this article.