The Inverse Trigonometric Functions Class 12 NCERT Solutions page compiles NCERT Class 12 Mathematics Chapter 2 into a download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, every formula matched to the CBSE marking scheme.
The solutions PDF covers all parts of every question and writes each step in full. The file is part of the NCERT Solutions Class 12 Maths library and follow the notation of the official textbook.
- CBSE Weightage: 4-6 marks (full Ch 2)
- JEE Main: 2-4% of paper
- Question Count in Ex 2.1: 14 (12 direct principal-value + 2 mixed/MCQ)
Student Pulse - Inverse Trigonometric Functions Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
How will Collegedunia's NCERT Solutions for Class 12 Maths Exercise 2.1 help you?
The Inverse Trigonometric Functions Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 2.1 looks deceptively simple - "find the principal value" - but most marks are lost at one specific step: picking the wrong branch. Our solutions force you to state the principal-value range first, then reduce the angle into that range, and only then write the answer. By Q5 you will stop confusing the cos-inverse range [0, π] with the sin-inverse range [-π/2, π/2].
For the negative-argument problems (questions 2, 4, 6, 8, 10), we apply the standard identities sin-1(-x) = -sin-1x and cos-1(-x) = π - cos-1x explicitly, instead of guessing the sign. That distinction is exactly what the 2026 marking scheme rewards with the second step mark.
Inverse Trigonometric Functions Exercise 2 1 Video Walkthrough
Source: Magnet Brains on YouTube
Principal Value Branches Used in Class 12 Maths Exercise 2.1
The exercise lives inside six fixed principal-value branches. Memorise the table below before attempting any question - it is the single most-tested fact in the Inverse Trigonometric Functions Class 12 NCERT Solutions and the only thing CBSE expects you to write as Step 1 of every answer.
| Function | Domain | Principal Value Branch (Range) | Questions in Ex 2.1 |
|---|---|---|---|
| sin-1x | [-1, 1] | [-π/2, π/2] | Q1, Q2, Q11 |
| cos-1x | [-1, 1] | [0, π] | Q3, Q4, Q9, Q12 |
| tan-1x | R | (-π/2, π/2) | Q7, Q8, Q13 |
| cosec-1x | R - (-1, 1) | [-π/2, π/2] - 0 | Q5, Q6 |
| sec-1x | R - (-1, 1) | [0, π] - π/2 | Q9, Q12 |
| cot-1x | R | (0, π) | Q10 |
Question-Wise Breakdown of NCERT Class 12 Maths Exercise 2.1
The Inverse Trigonometric Functions Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The 14 questions split into three difficulty tiers. The last two (Q13 and Q14) are MCQs that combine inverse functions; the rest are direct principal-value problems with one or two well-known reference angles.
| Q No. | Expression | Concept Tested | Principal Value | Difficulty |
|---|---|---|---|---|
| Q1 | sin-1(-1/2) | Negative sin-inverse | -π/6 | Easy |
| Q2 | cos-1(√3/2) | Standard cos-inverse | π/6 | Easy |
| Q3 | cosec-1(2) | Direct cosec-inverse | π/6 | Easy |
| Q4 | tan-1(-√3) | Negative tan-inverse | -π/3 | Easy |
| Q5 | cos-1(-1/2) | Negative cos-inverse; uses π - cos-1x identity | 2π/3 | Medium |
| Q6 | tan-1(-1) | Negative tan-inverse | -π/4 | Easy |
| Q7 | sec-1(2/√3) | Direct sec-inverse | π/6 | Medium |
| Q8 | cot-1(√3) | Direct cot-inverse | π/6 | Easy |
| Q9 | cos-1(-1/√2) | Negative cos-inverse | 3π/4 | Medium |
| Q10 | cosec-1(-√2) | Negative cosec-inverse | -π/4 | Medium |
| Q11 | tan-1(1) + cos-1(-1/2) + sin-1(-1/2) | Sum of three inverse values | 3π/4 | Hard |
| Q12 | cos-1(1/2) + 2sin-1(1/2) | Linear combination | 2π/3 | Hard |
| Q13 | MCQ: sin-1x = y gives | Range of sin-inverse | Option (B) -π/2 ≤ y ≤ π/2 | Easy |
| Q14 | MCQ: tan-1√3 - sec-1(-2) equals | Combined inverse-trig MCQ | Option (B) -π/3 | Hard |
Important Identities Used in Class 12 Maths Ex 2.1
The Inverse Trigonometric Functions Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Three negative-argument identities and the six reference-angle facts are the only things you must recall. Print this strip and keep it under the table during practice.
Negative argument: sin-1(-x) = -sin-1x , tan-1(-x) = -tan-1x , cosec-1(-x) = -cosec-1x .
Supplementary identity: cos-1(-x) = π - cos-1x , sec-1(-x) = π - sec-1x , cot-1(-x) = π - cot-1x .
Reference angles to memorise: sin(π/6) = 1/2 , sin(π/4) = 1/√2 , sin(π/3) = √3/2 , cos(π/6) = √3/2 , tan(π/4) = 1 , tan(π/3) = √3 .
Common pitfall: sin-1x ≠ (sin x)-1 . The first is an inverse function; the second is the reciprocal 1/sin x .
Sample Solved Question from Exercise 2.1
Here is Q5 solved in the exact Collegedunia step-format. Notice how the principal-value branch is stated first, then the reference angle, then the supplementary identity, and finally the answer.
Question 5: Find the principal value of cos-1(-12) .
Step 1 - State the principal value branch. The principal value branch of cos-1 is [0, π] .
Step 2 - Let the answer be y . Let y = cos-1(-1/2) . Then cos y = -1/2 , where y ∈ [0, π] .
Step 3 - Find the reference angle. We know cos(π/3) = 1/2 . So the reference angle is π/3 .
Step 4 - Use the supplementary identity. cos(π - π/3) = -cos(π/3) = -1/2 . Hence cos(2π/3) = -1/2 .
Step 5 - Verify the branch. Is 2π/3 ∈ [0, π] ? Yes. So y = 2π/3 .
Conclusion: The principal value of cos-1(-1/2) is 2π3 .
Common Mistakes Students Make in Class 12 Maths Ex 2.1
The Inverse Trigonometric Functions Class 12 NCERT Solutions are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
These six errors account for nearly every step-mark lost in Exercise 2.1 according to the CBSE 2025 marking-scheme commentary. Read them once before every revision sitting.
- Wrong branch picked. Writing cos-1(-1/2) = -π/3 is wrong - the cos-inverse range is [0, π], it cannot be negative. The correct answer is 2π/3 . This single error costs 2 marks per question.
- Confusing sin-1(-x) with cos-1(-x) . Sin-inverse and tan-inverse are odd functions; cos-inverse, sec-inverse, cot-inverse use the supplementary identity instead. Mixing them up is the #1 cause of wrong final values.
- Reading sin-1x as 1/sin x . The "-1" is functional notation, not an exponent. Misreading this turns the question into nonsense.
- Skipping Step 1. The first mark in each answer is for stating the principal value branch. Students work through the value and lose it.
- Forgetting the cosec/sec domain exclusions. cosec-1 is undefined on (-1, 1); sec-1 is undefined on (-1, 1). Always check the input lies in the domain first.
- Computing in degrees and forgetting to convert. NCERT answers must be in radians. Writing 30 instead of π/6 loses the final mark.
Related Resources for Class 12 Maths Chapter 2
- NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions (Full Chapter)
- Class 12 Maths Chapter 2 Notes
- Class 12 Maths Chapter 2 Formula Sheet
- Class 12 Maths Chapter 2 Exemplar Solutions
Inverse Trigonometric Functions Class 12 NCERT Solutions: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
Exercise-wise Breakdown of the Inverse Trigonometric Functions Chapter
The Inverse Trigonometric Functions chapter splits into 2 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 2.1 | Principal values of inverse trig functions |
| Exercise 2.2 | Properties and identities of inverse trig functions |
| Miscellaneous Exercise | Mixed inverse trig identities and equations |
PDF Download Formats and Languages for the Inverse Trigonometric Functions Chapter
The Inverse Trigonometric Functions Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The inverse trigonometric functions class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the inverse trigonometric functions class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The inverse trigonometric functions class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The inverse trigonometric functions class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The inverse trigonometric functions class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this inverse trigonometric functions class 12 pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the inverse trigonometric functions class 12 formulas pdf), and the full inverse trigonometric functions class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Inverse Trigonometric Functions Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Inverse Trigonometric Functions chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable inverse trigonometric functions class 12 important questions you will see on board day.
- inverse trigonometric functions class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The inverse trigonometric functions class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching inverse trigonometric functions class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Inverse Trigonometric Functions chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full inverse trigonometric functions class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Inverse Trigonometric Functions Notes Pair with NCERT Solutions and the Formula Sheet
The Inverse Trigonometric Functions Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Inverse Trigonometric Functions Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| inverse trigonometric functions class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| inverse trigonometric functions class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The inverse trigonometric functions class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The inverse trigonometric functions class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The inverse trigonometric functions class 12 formulas reference sheet is the same A4 file students sometimes refer to as inverse trigonometric functions class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board inverse trigonometric functions class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 inverse trigonometric functions solutions, class 12 inverse trigonometric functions ncert solutions, ncert class 12 inverse trigonometric functions solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Inverse Trigonometric Functions Class 12 |
|---|---|
| RD Sharma Class 12 Inverse Trigonometric Functions | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Inverse Trigonometric Functions | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo inverse trigonometric functions class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa inverse trigonometric functions class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board inverse trigonometric functions class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Inverse Trigonometric Functions | Advanced problems for JEE Main/JEE Advanced preparation. |
All NCERT Solutions for Inverse Trigonometric Functions Ex 2.1 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions Ex 2.1 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Find the principal value of sin-1(-12).
Find the principal value of cos-1(√32).
Find the principal value of -1(2).
Find the principal value of tan-1(-√3).
Find the principal value of cos-1(-12).
Find the principal value of tan-1(-1).
Find the principal value of sec-1(2√3).
Find the principal value of cot-1(√3).
Find the principal value of cos-1(-1√2).
Find the principal value of -1(-√2).
Find the value of tan-1(1) + cos-1(-12) + sin-1(-12).
Find the value of cos-1(12) + 2sin-1(12).
If sin-1 x = y, then
(A) 0 ≤ y ≤ π1.2em
(B) -π2 ≤ y ≤ π21.2em
(C) 0 < y < π1.2em
(D) -π2 < y < π2.
tan-1√3 - sec-1(-2) is equal to
(A) π1.2em(B) -π31.2em
(C) π31.2em(D) 2π3.
How to Use the Inverse Trigonometric Functions Notes Page Most Effectively
The recommended study plan for the Inverse Trigonometric Functions Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the inverse trigonometric functions class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The inverse trigonometric functions class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Inverse Trigonometric Functions Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 2.1 of Class 12 Maths Chapter 2?
Ans. Exercise 2.1 contains 14 questions in total: 12 direct principal-value problems on the six inverse trigonometric functions and 2 mixed/MCQ questions that combine inverse-trig values.
Ques. What is the main concept tested in Class 12 Maths Exercise 2.1?
Ans. The main concept is finding the principal value of an inverse trigonometric expression. Every answer must first state the principal-value branch, then reduce the input to a reference angle in that branch, then write the final value in radians.
Ques. What is the principal value branch of cos inverse and sin inverse?
Ans. The principal value branch of cos-1x is [0, π]; the principal value branch of sin-1x is [-π/2, π/2]. These are the most-tested branches in Ex 2.1 and must be quoted as Step 1 in every answer.
Ques. Why is cos -1 (-1/2) = 2π/3 and not -π/3 ?
Ans. Because cos-1 has range [0, π], the answer cannot be negative. Using the identity cos-1(-x) = π - cos-1x , we get cos-1(-1/2) = π - π/3 = 2π/3 , which lies in [0, π].
Ques. Is Exercise 2.1 part of the 2026-27 CBSE syllabus?
Ans. Yes. Inverse Trigonometric Functions remains a full chapter in the 2026-27 NCERT Class 12 Maths syllabus, and Exercise 2.1 on principal values is the foundation for the property-based simplifications in Exercise 2.2.
Ques. Where can I download the free PDF of NCERT Solutions for Class 12 Maths Exercise 2.1?
Ans. The PDF is available at the top of the Inverse Trigonometric Functions Class 12 NCERT Solutions. Click the download button to get the step-by-step solutions for all 14 questions of Exercise 2.1, prepared by Collegedunia subject experts as per the 2026-27 NCERT.
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