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CBSE Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions Worksheet with Answers

Q: What is the principal value branch of inverse trigonometric functions?

The principal value branch is the restricted range chosen for each inverse trigonometric function to make it a proper function (single-valued). For sin⁻¹x, it is [-π/2, π/2]; for cos⁻¹x, it is [0, π]; for tan⁻¹x, it is (-π/2, π/2). These ranges ensure that each input produces exactly one output.

Q: How many marks does Chapter 2 Inverse Trigonometric Functions carry in CBSE Class 12 board exams?

Inverse Trigonometric Functions typically carries 3-5 marks in the CBSE Class 12 Mathematics board exam. Questions usually appear as one 3-mark or 5-mark problem, or as part of integrated questions combining multiple chapters. The chapter is also important for calculus chapters that follow.

Q: Why does sin⁻¹(sin x) not always equal x?

Sin⁻¹(sin x) equals x only when x lies in the principal value branch [-π/2, π/2]. If x is outside this range, we must find the equivalent angle within [-π/2, π/2] that has the same sine value. For example, sin⁻¹(sin(3π/4)) = π/4, not 3π/4, because we use the reference angle.

Q: What is the most important formula for solving sum problems in inverse trigonometry?

The formula tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when xy < 1 is extremely important and frequently tested. Additionally, the complementary relations like sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2 are essential for simplifying expressions and solving equations quickly.

Q: How should I prepare Chapter 2 for scoring full marks in board exams?

Master all NCERT solved examples and complete exercises 2.1 and 2.2 thoroughly. Memorise domain and range of all six inverse functions. Practice proving identities systematically. Solve previous years' board papers to understand question patterns. Attempt worksheets like this one under timed conditions. Focus especially on composite function problems and equation-solving questions.

Q: What are the domain and range of sec⁻¹x and cosec⁻¹x?

For sec⁻¹x, domain is (-∞, -1] ∪ [1, ∞) and range is [0, π/2) ∪ (π/2, π]. For cosec⁻¹x, domain is (-∞, -1] ∪ [1, ∞) and range is [-π/2, 0) ∪ (0, π/2]. These exclude ±1 and certain angle values to maintain function properties.

Q: Can I use a calculator for inverse trigonometric function problems in CBSE exams?

Scientific calculators are generally not permitted in CBSE Class 12 Mathematics board exams. All inverse trigonometric problems are designed to be solved using standard angle values (0, π/6, π/4, π/3, π/2, etc.) and algebraic manipulation. Practice solving without a calculator to build confidence and speed.

Q: What is the difference between sin⁻¹x and (sin x)⁻¹?

Sin⁻¹x represents the inverse sine function (also written as arcsin x), which gives the angle whose sine is x. On the other hand, (sin x)⁻¹ means 1/(sin x), which equals cosec x. These are completely different operations — one is a functional inverse, the other is a reciprocal.

Q: How does CBSETUTOR.ai help with Inverse Trigonometric Functions problems?

CBSETUTOR.ai allows you to photograph any problem from this chapter and receive instant step-by-step solutions from an AI tutor trained on CBSE curriculum. Available 24×7, it explains domain restrictions, applies correct formulas, and shows detailed working for proofs and equations. At ₹999/month flat fee covering Classes 6-12 all subjects, with a 3-day free trial, it is affordable round-the-clock doubt support.

Q: Are there any shortcuts for solving inverse trigonometric equations quickly?

Yes, several: Memorise complementary pairs (sin⁻¹x + cos⁻¹x = π/2). For equations like sin⁻¹x + sin⁻¹y = constant, convert to sine of both sides. Use the tangent addition formula for sums of tan⁻¹ terms. Create right triangles to visualise composite functions like cos(sin⁻¹x). Always check domain validity before finalising answers.

Inverse Trigonometric Functions form a crucial part of CBSE Class 12 Mathematics, carrying significant weightage in board exams. This worksheet with answers provides structured practice across all difficulty levels, covering domain and range concepts, principal value determination, and important identities. Students should attempt this worksheet in one sitting under timed conditions to simulate exam pressure and build speed and accuracy.

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Key takeaways

✓Worksheet difficulty: Moderate to Challenging; Suggested time: 90 minutes for complete practice
✓Covers all essential topics from NCERT Class 12 Mathematics Chapter 2: domain, range, principal value, properties and identities of inverse trigonometric functions
✓Section-wise structure includes 6 MCQs, 5 fill-in-the-blanks, match-the-following, 5 short-answer questions (2-3 marks each), 3 long-answer HOTS questions (5 marks each), and 1 case-study problem
✓Answer key provides detailed explanations and working steps for every question to support self-learning
✓Questions designed to match CBSE board exam pattern with emphasis on principal value branches and composite function problems
✓Ideal for regular revision, pre-exam practice, and identifying weak areas in inverse trigonometric functions

Quick Chapter Recap: Inverse Trigonometric Functions

Before attempting the worksheet, quickly revise these core concepts from NCERT Class 12 Mathematics Chapter 2. Inverse trigonometric functions are the inverse functions of trigonometric ratios, restricted to specific domains to ensure they remain one-one and onto. The six inverse trigonometric functions are sin⁻¹x, cos⁻¹x, tan⁻¹x, cosec⁻¹x, sec⁻¹x, and cot⁻¹x. Each function has a defined domain and a corresponding range called the principal value branch. Understanding these principal value branches is essential for solving composite function problems. The chapter also covers important properties such as sin⁻¹(-x) = -sin⁻¹x, cos⁻¹(-x) = π - cos⁻¹x, tan⁻¹x + cot⁻¹x = π/2, and many more identities involving sum and composition of inverse trigonometric functions.

Domain of sin⁻¹x and cos⁻¹x: [-1, 1]; Range of sin⁻¹x: [-π/2, π/2]; Range of cos⁻¹x: [0, π]
Domain of tan⁻¹x and cot⁻¹x: all real numbers; Range of tan⁻¹x: (-π/2, π/2); Range of cot⁻¹x: (0, π)
Domain of sec⁻¹x and cosec⁻¹x: (-∞, -1] ∪ [1, ∞)
Key identity: sin⁻¹x + cos⁻¹x = π/2 for all x ∈ [-1, 1]
Important property: tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when xy < 1

Worksheet Difficulty Level and Time Guidelines

This CBSE Class 12 Mathematics Chapter 2 worksheet is designed at a moderate to challenging difficulty level, suitable for students who have completed the NCERT chapter and practised basic exercises. The worksheet comprises 25+ questions across six sections with varying complexity. Students should allocate approximately 90 minutes to complete the entire worksheet under exam-like conditions. Section A (MCQs) should take about 12 minutes, Section B (fill-in-the-blanks) around 10 minutes, Section C (match/true-false) about 8 minutes, Section D (short answers) approximately 25 minutes, Section E (long answers) around 30 minutes, and the case study about 10 minutes. Keep a calculator handy but remember that most CBSE board exam questions can be solved without one using standard values and algebraic manipulation.

Total suggested time: 90 minutes for complete worksheet completion
Difficulty progression: begins with concept-based MCQs, advances to application-based long answers
Marking scheme follows CBSE pattern: MCQs (1 mark), Fill-ups (1 mark), Short answers (2-3 marks), Long answers (5 marks)
Calculator usage: optional but not necessary for most questions
Recommended for students who have finished NCERT exercises 2.1 and 2.2

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing fundamental concepts of inverse trigonometric functions. Each question carries one mark. Choose the correct option from the four alternatives provided. These questions cover domain and range identification, principal value calculation, and basic property application. Carefully read each question, eliminate obviously incorrect options, and select the most appropriate answer. Remember that negative marking is typically not applied in CBSE board exams, so attempt all MCQs even if slightly unsure.

Section B: Fill in the Blanks (1 mark each)

Complete each statement by filling in the blank with the correct mathematical expression, value, or term. These questions test your recall of important formulas, domain-range pairs, and standard results from NCERT Class 12 Mathematics Chapter 2. Write your answers clearly and include proper mathematical notation where required. Pay attention to exact values using π notation rather than decimal approximations unless specifically asked otherwise. Each correct answer carries one mark with no partial credit typically awarded for fill-in-the-blank questions in CBSE examinations.

Section C: Match the Following and True/False (1 mark each)

This section contains two sub-parts. Part I requires you to match items from Column A with their corresponding entries in Column B. Write the correct pairs clearly. Part II contains true/false statements about inverse trigonometric functions. Mark each statement as True (T) or False (F). For false statements, be prepared to justify why they are incorrect during self-review. These questions assess your understanding of properties, identities, and domain-range concepts. Each correct match or true/false identification carries one mark. Accuracy is crucial as these are straightforward concept-checking questions that should not consume much time during the exam.

Section D: Short Answer Questions (2-3 marks each)

Attempt these five short-answer questions showing all necessary working steps. Each question carries 2 to 3 marks depending on complexity. Marks are awarded for correct method, proper use of formulas, and accurate final answers. Write clearly, showing intermediate steps so that partial credit can be awarded even if the final answer contains a minor error. These questions typically test your ability to apply properties of inverse trigonometric functions, solve equations involving inverse functions, and simplify composite expressions. Budget approximately 4-5 minutes per question in this section. Use standard trigonometric identities and inverse function properties covered in NCERT Class 12 Mathematics Chapter 2.

Section E: Long Answer and HOTS Questions (5 marks each)

This section contains three long-answer questions designed to test higher-order thinking skills and deeper understanding of inverse trigonometric functions. Each question carries 5 marks and requires detailed working with proper justification. These problems often combine multiple concepts and may involve proving identities, solving complex equations, or applying inverse trigonometric functions in novel contexts. Allocate about 10 minutes per question. Full marks require correct mathematical reasoning, proper use of domain restrictions, accurate algebraic manipulation, and a clearly stated final answer. Show all steps as significant partial credit is available even if you cannot reach the final answer. These questions reflect the difficulty level of 5-mark questions in CBSE Class 12 board examinations.

Section F: Case Study Based Question (4 marks)

Case-study questions have become an integral part of CBSE Class 12 Mathematics board exams since 2020-21. This section presents a real-world scenario or theoretical context followed by 3-4 sub-questions worth a total of 4 marks. Read the case study carefully, understand the given information, and apply concepts from Inverse Trigonometric Functions to answer the questions. These questions test your ability to extract mathematical problems from descriptive contexts and apply chapter concepts in practical situations. The case study typically includes 2-3 sub-questions of 1 mark each and 1 sub-question of 2 marks. Show working for calculation-based parts and write clear, concise answers for conceptual parts. This format mirrors the actual CBSE board exam pattern and helps students prepare for application-based assessment.

Complete Answer Key with Explanations

Below is the comprehensive answer key for all sections of this CBSE Class 12 Mathematics Chapter 2 worksheet. Each answer includes brief working steps or explanations to help you understand the solution method. Cross-check your answers carefully, award yourself marks honestly, and identify topics that need more practice. If you score below 60% in any section, revise that topic from NCERT Class 12 Mathematics textbook and attempt additional problems from the exercise. For conceptual doubts, consider using CBSETUTOR.ai where you can upload problem photos and get instant step-by-step solutions from an AI tutor available 24×7, all at a flat fee of ₹999 per month with a 3-day free trial. Review incorrect answers thoroughly to understand your mistakes and avoid repeating them in exams.

Section A Answers: Q1(b), Q2(b), Q3(a), Q4(b), Q5(b), Q6(b)
Section B Answers: Q7: [0, π/2) ∪ (π/2, π], Q8: 1/2, Q9: 5π/6, Q10: -sin⁻¹x, Q11: [-1, 1], Q12: a and b can be any real numbers (assuming standard domain)
Section C Answers: Q13-b, Q14-a, Q15-c, Q16-d; Q17-True, Q18-False, Q19-False, Q20-True
Total marks: 44 (Section A: 6, Section B: 6, Section C: 8, Section D: 13, Section E: 15, Section F: 4)

How to Use This Worksheet Effectively for Board Exam Preparation

This CBSE Class 12 Mathematics Chapter 2 worksheet is a comprehensive practice tool when used strategically. First, ensure you have completed all NCERT exercises from the chapter before attempting this worksheet. Set a timer for 90 minutes and attempt the worksheet in one sitting without referring to notes or solutions, simulating actual exam conditions. After completion, use the answer key to evaluate your performance section-wise. Calculate your score in each section to identify weak areas. For topics where you scored poorly, return to NCERT Class 12 Mathematics textbook, re-read the relevant theory, and solve additional problems. Attempt the worksheet again after a week to measure improvement. If specific question types consistently trouble you, CBSETUTOR.ai offers personalised practice where you can upload doubt photos and receive instant solutions with step-by-step explanations from an AI tutor, available round-the-clock at just ₹999 per month covering all subjects for Classes 6-12, with a risk-free 3-day trial.

First attempt: closed-book, timed at 90 minutes to simulate exam pressure
Score yourself honestly using the answer key; identify sections scoring below 70%
Revisit NCERT theory and examples for weak topics, then re-solve those worksheet sections
Second attempt after one week: aim for 20% improvement in overall score
Focus on understanding solution methods, not just memorising answers
Use error analysis: categorise mistakes as conceptual, calculation-based, or time-management issues

Common Mistakes to Avoid in Inverse Trigonometric Functions

Students frequently make predictable errors when solving problems from CBSE Class 12 Mathematics Chapter 2. The most common mistake is forgetting domain restrictions, especially when simplifying expressions like sin⁻¹(sin x) or cos⁻¹(cos x), which equal x only when x lies within the principal value branch. Another frequent error is incorrect application of the formula tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)], which holds only when xy < 1; when xy > 1, we must add or subtract π depending on the signs of x and y. Students also confuse the ranges of different inverse functions, leading to wrong quadrant identification. Sign errors are common in properties like sin⁻¹(-x) = -sin⁻¹x versus cos⁻¹(-x) = π - cos⁻¹x. When solving equations, always verify that your final answer lies within the valid domain of the variable. During exams, double-check angle measurements, use exact values (π/6, π/4, etc.) rather than decimal approximations, and clearly state the principal value branch when answering questions about range.

Domain neglect: forgetting that sin⁻¹(sin(5π/6)) ≠ 5π/6 because 5π/6 is outside [-π/2, π/2]
Formula misapplication: using tan⁻¹x + tan⁻¹y formula without checking the condition xy < 1
Range confusion: stating range of tan⁻¹x as [0, π] instead of correct (-π/2, π/2)
Sign errors: writing cos⁻¹(-x) = -cos⁻¹x, which is incorrect
Calculation mistakes: errors in simplifying expressions like √(1 - x²) within composite functions
Not verifying solutions against domain constraints after solving equations

Additional Resources and CBSETUTOR.ai Support

Beyond this worksheet, students preparing for CBSE Class 12 Mathematics board exams should utilise multiple resources. The NCERT textbook remains the primary resource, with all theoretical explanations, solved examples, and chapter exercises. NCERT Exemplar provides additional challenging problems ideal for competitive exam preparation. Previous years' CBSE board papers show actual exam patterns and frequently tested question types. Sample papers released by CBSE should be practised under timed conditions. For students seeking personalised help, CBSETUOR.ai offers an innovative solution where you can photograph any problem and receive instant, detailed solutions from an AI tutor trained specifically on CBSE curriculum. Whether it is midnight doubt-clearing before an exam or regular practice support, CBSETUTOR.ai provides 24×7 assistance covering all chapters of Class 12 Mathematics at a nominal flat fee of ₹999 per month for unlimited questions across all subjects from Class 6 to 12. Start with a 3-day free trial to experience how AI-powered tutoring can supplement your preparation.

NCERT Class 12 Mathematics textbook: master all solved examples and unsolved exercises
NCERT Exemplar: attempt MCQs and long-answer questions for deeper conceptual clarity
Previous 5 years' CBSE board papers: identify repeated question patterns and frequently tested concepts
CBSE sample papers: solve under timed conditions, one per week leading up to exams
CBSETUTOR.ai: upload problem photos anytime for instant step-by-step solutions from AI tutor; ₹999/month flat fee, 3-day free trial, covers all subjects Classes 6-12

Frequently asked questions

What is the principal value branch of inverse trigonometric functions?+

The principal value branch is the restricted range chosen for each inverse trigonometric function to make it a proper function (single-valued). For sin⁻¹x, it is [-π/2, π/2]; for cos⁻¹x, it is [0, π]; for tan⁻¹x, it is (-π/2, π/2). These ranges ensure that each input produces exactly one output.

How many marks does Chapter 2 Inverse Trigonometric Functions carry in CBSE Class 12 board exams?+

Inverse Trigonometric Functions typically carries 3-5 marks in the CBSE Class 12 Mathematics board exam. Questions usually appear as one 3-mark or 5-mark problem, or as part of integrated questions combining multiple chapters. The chapter is also important for calculus chapters that follow.

Why does sin⁻¹(sin x) not always equal x?+

Sin⁻¹(sin x) equals x only when x lies in the principal value branch [-π/2, π/2]. If x is outside this range, we must find the equivalent angle within [-π/2, π/2] that has the same sine value. For example, sin⁻¹(sin(3π/4)) = π/4, not 3π/4, because we use the reference angle.

What is the most important formula for solving sum problems in inverse trigonometry?+

The formula tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when xy < 1 is extremely important and frequently tested. Additionally, the complementary relations like sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2 are essential for simplifying expressions and solving equations quickly.

How should I prepare Chapter 2 for scoring full marks in board exams?+

Master all NCERT solved examples and complete exercises 2.1 and 2.2 thoroughly. Memorise domain and range of all six inverse functions. Practice proving identities systematically. Solve previous years' board papers to understand question patterns. Attempt worksheets like this one under timed conditions. Focus especially on composite function problems and equation-solving questions.

What are the domain and range of sec⁻¹x and cosec⁻¹x?+

For sec⁻¹x, domain is (-∞, -1] ∪ [1, ∞) and range is [0, π/2) ∪ (π/2, π]. For cosec⁻¹x, domain is (-∞, -1] ∪ [1, ∞) and range is [-π/2, 0) ∪ (0, π/2]. These exclude ±1 and certain angle values to maintain function properties.

Can I use a calculator for inverse trigonometric function problems in CBSE exams?+

Scientific calculators are generally not permitted in CBSE Class 12 Mathematics board exams. All inverse trigonometric problems are designed to be solved using standard angle values (0, π/6, π/4, π/3, π/2, etc.) and algebraic manipulation. Practice solving without a calculator to build confidence and speed.

What is the difference between sin⁻¹x and (sin x)⁻¹?+

Sin⁻¹x represents the inverse sine function (also written as arcsin x), which gives the angle whose sine is x. On the other hand, (sin x)⁻¹ means 1/(sin x), which equals cosec x. These are completely different operations — one is a functional inverse, the other is a reciprocal.

How does CBSETUTOR.ai help with Inverse Trigonometric Functions problems?+

CBSETUTOR.ai allows you to photograph any problem from this chapter and receive instant step-by-step solutions from an AI tutor trained on CBSE curriculum. Available 24×7, it explains domain restrictions, applies correct formulas, and shows detailed working for proofs and equations. At ₹999/month flat fee covering Classes 6-12 all subjects, with a 3-day free trial, it is affordable round-the-clock doubt support.

Are there any shortcuts for solving inverse trigonometric equations quickly?+

Yes, several: Memorise complementary pairs (sin⁻¹x + cos⁻¹x = π/2). For equations like sin⁻¹x + sin⁻¹y = constant, convert to sine of both sides. Use the tangent addition formula for sums of tan⁻¹ terms. Create right triangles to visualise composite functions like cos(sin⁻¹x). Always check domain validity before finalising answers.

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