CBSE Class 12 Maths Revision Notes Chapter 2

Class 12 Mathematics Chapter 2 Notes

Mathematics is a subject that demands lots of practice. Students need to have an in-depth knowledge of all basic concepts to understand more complex topics of the subject. The Class 12 Mathematics Chapter 2 notes introduce various concepts of the inverse of trigonometric functions. This chapter is very important and has high weightage in the board examinations. These concepts are also to be used to solve MCQs asked in JEE Mains and other national-level competitive examinations. 

Extramarks provides stepwise and detailed Class 12 Mathematics Chapter 2 notes, keeping a note on its overall importance. It ensures sufficient practice through problem-solving and helps to clear doubts and queries. 

Students should stay tuned with Extramarks to update themselves with all the latest information regarding CBSE exams, syllabus, paper patterns, and more. 

Key Topics Covered In Class 12 Mathematics Chapter 2 Notes

The key topics covered in class 12 Mathematics chapter 2 notes include the following.

Inverse function:

Let x and y be two functions such that y = f(x) and x= g(y). Composition of function fog(y) = f(g(y)) and gof(y) = g(f(y))=x then f and y are invertible or inverse of each other, i.e., g = f-1. We can say that, if y=f(x) then x = f-1(y). 

Inverse trigonometric function: 

If y = f(x) = sin x then its inverse x = sin-1(y), where y ∈ [–2 , 2 ] and x ∈ [-1,1]. This is applicable for all trigonometric functions. 

Domain, Range and Graphs of all Inverse trigonometric functions:

Graphs of Inverse Trigonometric functions:

1.  y= sin-1(x)

2. y= cos-1(x)

3. y = tan-1(x)

4. y= cot-1(x)

5. y = cosec-1(x)

6. y= sec-1(x)

NOTE: 

1. sin-1(x) and tan-1(x) are increasing functions, whereas cos-1(x) and cot-1(x) are decreasing functions over their domain. 
2. sin-1(x)  and (sin (x))-1 are different and should not be confused. 

Properties: 

Property 1: Relation between two trigonometric functions. 

1. sin -1(1x)= cosec-1(x) values of x ∈(−∞,1] ∪ [1,∞)
2. cos -1(1x)= sec-1(x) values of x ∈ (−∞,1] ∪ [1,∞)
3. tan -1(1x)= cot-1(x) values of x > 0

                   = – + cot-1(x) values of x < 0

Property 2: Negative angle Inverse Trigonometric identities

1. sin -1(x)= -sin-1(x) values of x ∈ [-1, 1]
2. tan -1(x)= -tan-1(x) values of x ∈ R
3. cosec -1(x)= -cosec-1(x) values of x ∈ (−∞,−1]∪[1,∞)
4. cos -1(x)= -cos-1(x) values of x ∈ [-1, 1]
5. sec -1(x)= -sec-1(x) values of x ∈ (−∞,−1]∪[1,∞)
6. cot -1(x)= -cot-1(x) values of x ∈ R

Property 3:

1. sin -1(x)+ cos -1(x)= 2 values of x ∈ [-1, 1]
2. tan -1(x)+ cot -1(x)= 2 values of x ∈ R
3. sec -1(x)+ cosec -1(x)= 2 values of x ∈ (−∞,−1]∪[1,∞)

Property 4: 

1. tan -1(x) + tan -1(y)= tan -1x + y1-xy values of xy < 1
2. tan -1(x) – tan -1(y)= tan -1x – y1+xy values of xy > -1
3. tan -1(x) – tan -1(y)= + tan -1(x + y1-xy) values of xy > 1; x,y= 0

Property 5:

1. 2tan -1(x)= sin -1(2x1+x2), x 1
2. 2tan -1(x)= cos -1(1-x21+x2), x 0
3. 2tan -1(x)= tan -1(2x1-x2), -1x 1

Property 6:

1. sin(sin -1(x)) = x, –2 x2

                          = -x, 2 x32

1. cos(cos -1(x)) = x, 0 x

                          = 2-x, x 2

1. tan(tan -1(x)) = –-x, x [–32 ,- 2 ] 

                           = x, x [–2 , 2 ]

                           = x – , x [2 ,32 ]

                           = x – 2, x [32 ,52 ]

Additional Formulas: 

1. sin -1(x) + sin -1(y) = sin -1(x1- y2 +y1- x2   )
2. sin -1(x) – sin -1(y) = sin -1(x1- y2 -y1- x2   )
3. cos -1(x) + cos -1(y) = cos -1(xy –1- y2 1- x2   )
4. cos -1(x) – cos -1(y) = cos -1(xy +1- y2 1- x2   )
5. tan -1(x) + tan -1(y) + tan -1(z)= tan -1x + y + z – xyz1 – xy – yz – xz , if x, y, z>0 & xy+ yz+ zx<1
6. If tan -1(x) + tan -1(y) + tan -1(z)= , then x+ y+ z= xyz
7. If tan -1(x) + tan -1(y) + tan -1(z)= 2, then xy+ yz+ z
8. sin -1(x) + sin -1(y)+ sin -1(z)= 32, x= y= z= 1
9. cos -1(x) + cos -1(y)+ cos -1(z)= 3, x= y= z= -1

Remember:

• When approaching the equations, square them so that it becomes easier to solve. Sometimes the answers or roots of the equation will not satisfy the original equation.
• Do not cancel common factors involving unknown angles on the left-hand side (LHS) and righthand side (RHS) of an equation because it may be the solution of the given equation.
• The solution of any equation, including sec θ or tan θ, can never be in the form (2n + 1) π / 2.
• The solution of any equation, including cosec θ or cot θ, can never be in the form θ = nπ. 

Class 12 Mathematics Chapter 2 Notes Exercises & Answer Solutions.

Chapter 2 of mathematics class 12 helps students to learn the inverse trigonometric functions, their range domain and graphs. It is an important chapter and holds relevance in the 12th board as well as in various entrance exams like JEE, State Engineering Entrance Exams, etc. Extramarks has prepared class 12 mathematics chapter 2 notes which include all Exercise and Answer Solutions. Students can refer to the solutions provided for exercise questions and seek help if they face any difficulties or doubts. The CBSE revision notes include detailed information about theorems, formulas, and derivations in a very descriptive way. 

The class 12 mathematics chapter 2 notes are prepared by experts with the objective to provide in-depth knowledge of all the concepts. Practising questions with the help of these solutions will also help to develop analytical and problem-solving skills. 

For better results in the examinations, students may refer to the Extramarks Class 12 Mathematics Chapter 2 notes along with MCQ questions of this chapter from the links below.

Extramarks, with the right guidance and support, empower and provides a holistic learning experience to learners. The class 12 mathematics notes chapter 2 guides students to give their best in preparing for the examinations. Experience the power of wholesome learning with Extramarks! 

NCERT Exemplar Class 12 Mathematics

Keeping the overall importance of the subject in our mind, Extramarks, an online learning platform, provides academic notes such as NCERT Exemplar, Class 12 Mathematics Chapter 2 notes, and more. The notes are prepared by teachers and experts in a descriptive manner to enable students to improve the learning process. With the help of the notes, students can clear their queries and develop in-depth knowledge of the chapter. The Class 12 Mathematics Chapter 2 notes have solutions to all answers present in the NCERT books. It allows a sufficient amount of practice and improves problem-solving skills. NCERT Exemplar Books for class 12 is one of the best resources for efficient exam preparation. 

Key Features of Class 12 Mathematics Chapter 2 Notes

The key features of Class 12 Mathematics Chapter 2 notes provided by Extramarks include

• The notes are prepared after extensive research by the subject elites at Extramarks.
• It is based on the latest CBSE syllabus and follows all the revised guidelines issued by the board.
• The concepts included in the notes are explained lucidly in easy language. 
• The class 12 chapter 2 mathematics notes offer students many CBSE extra questions so that they can get acquainted with the different types of questions that are asked in the examinations.
• It reduces stress and boosts confidence during your exam preparation.
• With the help of the notes, students learn about the important details of CBSE exams, such as exam paper patterns, marking schemes, weightage of each chapter, etc. 
• These notes are a beneficial resource that will help students to know the chapter’s most important concepts and weightage to plan their studies.
• Students learn to manage time well and develop analytical and problem-solving skills, thus reducing the chances of mistakes. 

To get the best learning experience through conceptual clarity, visit the Extramarks website. The academic material, such as the Class 12 Mathematics Chapter 2 notes, provides end-to-end learning solutions. Also, get access to various CBSE sample papers and CBSE previous year question papers on the Extramarks web portal.