This page hosts the Inverse Trigonometric Functions Formulas, and hosts the Inverse Trigonometric Functions Formulas for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions. The file presents the formula sheet in the order they appear in the NCERT textbook, with each entry followed by a one-line note on its standard use. the PDF are intended for last-revision and exam-day recall.
6 pages · complete sheet
24 formulas · full chapter
6 functions · principal branches
- CBSE Weightage: 4 to 5 marks (typically one 2-mark simplification plus one MCQ on principal value).
- JEE Main Weightage: 3 to 5% of the Maths section (1 to 2 questions per shift).
- CUET (UG) Weightage: 1 to 2 MCQs in nearly every shift.
Chapter 2 Inverse Trigonometric Functions Formula Sheet PDF
Complete Inverse Trigonometric Functions Formulas with Principal-Value Branches
| Concept | Formula / Rule | NCERT Section | Common Use |
|---|---|---|---|
| Principal range of sin-1 x | [-π2, π2] , domain [-1, 1] | 2.2 | Sign decision in every Q |
| Principal range of cos-1 x | [0, π] , domain [-1, 1] | 2.2 | Sign decision in every Q |
| Principal range of tan-1 x | (-π2, π2) , domain R | 2.2 | Sign decision in every Q |
| Principal range of cot-1 x | (0, π) , domain R | 2.2 | Sign decision in every Q |
| Principal range of sec-1 x | [0, π] π2 , domain |x| ≥ 1 | 2.2 | Domain check |
| Principal range of csc-1 x | [-π2, π2] 0 , domain |x| ≥ 1 | 2.2 | Domain check |
| Negative-argument (sin) | sin-1(-x) = -sin-1 x | 2.3 | Sign flipping |
| Negative-argument (cos) | cos-1(-x) = π - cos-1 x | 2.3 | Sign flipping |
| Negative-argument (tan) | tan-1(-x) = -tan-1 x | 2.3 | Sign flipping |
| Negative-argument (cot) | cot-1(-x) = π - cot-1 x | 2.3 | Sign flipping |
| Reciprocal identity (sin) | csc-1 x = sin-1(1/x) , |x| ≥ 1 | 2.3 | Converting csc to sin |
| Reciprocal identity (cos) | sec-1 x = cos-1(1/x) , |x| ≥ 1 | 2.3 | Converting sec to cos |
| Reciprocal identity (tan) | cot-1 x = tan-1(1/x) , x > 0 | 2.3 | Converting cot to tan |
| Complementary sin-cos | sin-1 x + cos-1 x = π2 , x ∈ [-1, 1] | 2.3 | 2-mark CBSE |
| Complementary tan-cot | tan-1 x + cot-1 x = π2 , x ∈ R | 2.3 | 2-mark CBSE |
| Complementary sec-csc | sec-1 x + csc-1 x = π2 , |x| ≥ 1 | 2.3 | 1-mark MCQ |
| Sum formula (tan) | tan-1 x + tan-1 y = tan-1(x+y1-xy) , xy < 1 | 2.3 (JEE extension) | JEE Main |
| Difference formula (tan) | tan-1 x - tan-1 y = tan-1(x-y1+xy) , xy > -1 | 2.3 (JEE extension) | JEE Main |
| Double-angle (tan to sin) | 2tan-1 x = sin-1(2x1+x2) , |x| ≤ 1 | 2.3 (JEE extension) | JEE Main |
| Double-angle (tan to cos) | 2tan-1 x = cos-1(1-x21+x2) , x ≥ 0 | 2.3 (JEE extension) | JEE Main |
| Double-angle (tan to tan) | 2tan-1 x = tan-1(2x1-x2) , |x| < 1 | 2.3 (JEE extension) | JEE Main |
| sin-1 x in terms of cos | sin-1 x = cos-1√1-x2 , x ∈ [0, 1] | 2.3 | Right-triangle conversion |
| sin-1 x in terms of tan | sin-1 x = tan-1(x√1-x2) , x ∈ (-1, 1) | 2.3 | Right-triangle conversion |
| Composite inverse identities | sin(sin-1 x) = x on [-1,1]; sin-1(sin x) = x on [-π2, π2] | 2.2 | 1-mark MCQ |
Inverse Trigonometric Functions Video Walkthrough
Source: Magnet Brains on YouTube
Principal-Value Branch Quick-Fact Cards for MCQ Recall
sin-1
Range [-π2, π2], domain [-1, 1].
cos-1
Range [0, π], domain [-1, 1].
tan-1
Range (-π2, π2), domain R.
cot-1
Range (0, π), domain R.
sec-1
Range [0, π] π2, domain |x| ≥ 1.
csc-1
Range [-π2, π2] 0, domain |x| ≥ 1.
How the Inverse Trigonometric Functions Formulas on the Inverse Trigonometric Functions Formulas Help You
- 2026-27 NCERT Alignment: Every formula matches the current edition; JEE-only sum, difference, and double-angle rules are clearly flagged as extension rows so board-only students can skip them.
- Principal-Value Shortcuts: The six ranges are formatted as quick-fact cards because deciding the correct branch is the single highest-use 1-mark question of the Inverse Trigonometric Functions Formulas.
- Complementary-Pair Recall: The three π2 pair identities sit on consecutive rows; CBSE 2-mark questions almost always test one of these.
- Triangle-Conversion Map: The rows that translate sin-1 x into cos-1 or tan-1 cover the right-triangle simplification setup CBSE has asked four of the last five years.
When to Use Which Formula in Inverse Trigonometric Functions
- Question asks "principal value of sin-1(sin x) ": reduce x to the range [-π2, π2] using sin(π - x) = sin x or sin(2π + x) = sin x ; the answer is the reduced value, not x itself.
- Question simplifies sin-1 x + cos-1 x : apply sin-1 x + cos-1 x = π2 immediately; do not expand into sin and cos.
- Question simplifies tan-1 x + tan-1 y : check the sign of xy first. If xy < 1, use the direct sum; if xy > 1 and both are positive, add π to the formula's output.
- Question converts 2tan-1 x : the result depends on the range of x: sin form for |x| ≤ 1, cos form for x ≥ 0, tan form for |x| < 1. Pick by what the answer choices look like.
- Question asks "draw the graph": sketch using the principal-value range as the y-axis ceiling and floor; sin-1 x and tan-1 x are monotonically increasing, cos-1 x and cot-1 x are decreasing.
- Question gives a right-triangle setup: convert the inverse function into a triangle (opposite, adjacent, hypotenuse) and read off the required ratio.
Quick Tip: The output of sin-1(sin x) is NOT always x. It equals x only when x ∈ [-π2, π2]. For x = 2π3, the answer is π3 (since sin(2π3) = sin(π3) and π3 is in the principal range). This single trap has appeared on CBSE papers in 2022 and 2024.
CBSE and JEE Numerical Pattern Templates for Inverse Trigonometric Functions
| Template | Typical Setup | Formula to Reach For |
|---|---|---|
| Principal-value evaluation | "Find the principal value of tan-1(-√3) ." | Range of tan-1 = (-π2, π2); answer -π3 |
| Complementary simplification | "Simplify sin-1(0.6) + cos-1(0.6) ." | sin-1 x + cos-1 x = π2 |
| Sum of two tan inverses | "Find tan-1(1/2) + tan-1(1/3) ." | Tan sum formula; xy = 1/6 < 1; answer π4 |
| Double-angle conversion | "Express 2tan-1(13) as a single sin inverse." | 2tan-1 x = sin-1(2x1+x2) ; answer sin-1(35) |
Common Mistakes Students Make in Inverse Trigonometric Functions
Watch Out: sin-1(sin x) ≠ x outside the principal range. Always reduce x into [-π2, π2] before writing the answer.
Watch Out: cos-1(-x) = π - cos-1 x , not -cos-1 x. The cos and cot families flip with π - θ; only sin, tan, and csc inverses are pure odd.
Watch Out: The tan sum formula tan-1 x + tan-1 y = tan-1(x+y1-xy) is only valid when xy < 1. For xy > 1 with positive x, y, add π ; for xy > 1 with negative x, y, subtract π .
Watch Out: sec-1 x is NOT defined for |x| < 1. The domain restriction |x| ≥ 1 appears in CBSE assertion-reason MCQs almost every year.
Memory Mnemonics for Inverse Trigonometric Functions
Remember: "Sin and Tan live below the line, Cos and Cot live above." Sin, tan, csc inverses use the symmetric range around zero (-π2 to π2 ); cos, cot, sec inverses use the upper range (0 to π ).
Remember: "Pi-over-two pairs". The three identities sin-1 + cos-1 , tan-1 + cot-1 , sec-1 + csc-1 all sum to π2 . Each is a 1-mark MCQ ready to happen.
Remember: Negative goes inside for sin, tan, csc (odd functions); negative becomes π - for cos, cot, sec. Use the phrase "STC odd, CCS pi-minus" to lock it in.
Top 5 Most-Asked Inverse Trigonometric Functions Topics in Class 12 Board Exams
- Principal-value evaluation of sin-1(sin x) or cos-1(cos x) with x outside the principal range (2 marks, almost every year).
- Simplification using complementary pairs like sin-1 x + cos-1 x = π2 (1-mark MCQ, every year).
- Sum or difference of two tan-1 terms with a check on xy < 1 (2 marks, recurring).
- Right-triangle conversion such as sin-1 x = tan-1(x√1-x2) (2-mark short-answer).
- Graphical recognition of the six inverse functions and their principal ranges (assertion-reason MCQ).
More Inverse Trigonometric Functions Class 12 Maths Resources
- Inverse Trigonometric Functions Class 12 Maths NCERT Solutions
- Inverse Trigonometric Functions Class 12 Maths Notes
- Inverse Trigonometric Functions Class 12 Maths NCERT Book PDF
- Inverse Trigonometric Functions Class 12 Maths NCERT Exemplar Book PDF
- Inverse Trigonometric Functions Class 12 Maths NCERT Exemplar Solutions
- Inverse Trigonometric Functions Class 12 Maths Handwritten Notes
NCERT Formula Sheet for Class 12 Maths: All Chapters
| Chapter | Resource |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Exercise-wise Breakdown of the Inverse Trigonometric Functions Chapter
| 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
| 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 |
- 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.
Important Questions and Previous Year Trends for the Inverse Trigonometric Functions Chapter
| 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. |
- 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
| 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 |
How the Inverse Trigonometric Functions Notes Pair with NCERT Solutions and the Formula Sheet
| 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 |
- 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
| 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. |
How to Use the Inverse Trigonometric Functions Notes Page Most Effectively
| 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. |
- 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.
This Collegedunia NCERT Class 12 Mathematics page is reviewed against every CBSE board paper release.
The Collegedunia Class 12 Maths team checks every formula on this page against the printed NCERT 2026-27 edition.
Inverse Trigonometric Functions Formulas - Frequently Asked Questions
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