Inverse Trigonometric Functions (ITF) Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Inverse Trigonometric Functions — Competishun
Inverse Trigonometric Functions (ITF) Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Inverse Trigonometric Functions (ITF) is one of the most important chapters in Class 12 Maths for JEE and board exams. It carries moderate weightage in JEE Main and is a guaranteed source of marks. The chapter deals with the inverse operations of trigonometric functions, which are essential for solving many problems in calculus, coordinate geometry, and algebra.
This chapter covers the definition of inverse trigonometric functions, their domain and range, principal value branches, important properties, graphs, and formulas. Understanding these concepts is essential for solving complex problems in JEE and beyond.
This page gives you the complete guide to Inverse Trigonometric Functions with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.
Download the Inverse Trigonometric Functions Complete Guide PDF
Get all Inverse Trigonometric Functions concepts, domain and range, principal values, properties, formulas, and practice questions in one clean PDF, free. Perfect for JEE and board revision.
Download Free PDFWhat are Inverse Trigonometric Functions?
The main inverse trigonometric functions are:
sin⁻¹x
Inverse of sine function. Returns the angle whose sine is x.
cos⁻¹x
Inverse of cosine function. Returns the angle whose cosine is x.
tan⁻¹x
Inverse of tangent function. Returns the angle whose tangent is x.
Glossary of ITF Terms — Complete A to Z
Before diving deep into each topic, let's understand the key terminology used in this chapter:
| Term | Definition |
|---|---|
| Inverse Trigonometric Function | The inverse of a trigonometric function, denoted by sin⁻¹x, cos⁻¹x, etc. |
| Principal Value Branch | The restricted range of an inverse trigonometric function that makes it single-valued. |
| Domain | The set of all possible input values (x) for which the function is defined. |
| Range | The set of all possible output values (the principal values). |
| sin⁻¹x (arcsin x) | The inverse sine function. Domain: [-1, 1], Range: [-π/2, π/2]. |
| cos⁻¹x (arccos x) | The inverse cosine function. Domain: [-1, 1], Range: [0, π]. |
| tan⁻¹x (arctan x) | The inverse tangent function. Domain: R, Range: (-π/2, π/2). |
| cot⁻¹x (arccot x) | The inverse cotangent function. Domain: R, Range: (0, π). |
| sec⁻¹x (arcsec x) | The inverse secant function. Domain: (-∞, -1] ∪ [1, ∞), Range: [0, π] \ {π/2}. |
| cosec⁻¹x (arccosec x) | The inverse cosecant function. Domain: (-∞, -1] ∪ [1, ∞), Range: [-π/2, π/2] \ {0}. |
| Mastering these terms is essential for understanding Inverse Trigonometric Functions. | |
Domain, Range, and Principal Value Branches
Important Properties of Inverse Trigonometric Functions
1. Negative Arguments
sin⁻¹(−x)
tan⁻¹(−x)
cos⁻¹(−x)
cot⁻¹(−x)
2. Complementary Pairs (Sum = π/2)
sin⁻¹x + cos⁻¹x
tan⁻¹x + cot⁻¹x
sec⁻¹x + cosec⁻¹x
3. Reciprocal Arguments
sin⁻¹(1/x)
cos⁻¹(1/x)
tan⁻¹(1/x)
4. Conversion Formulas
sin⁻¹x
cos⁻¹x
tan⁻¹x
tan⁻¹x
Addition and Subtraction Formulas for ITF
tan⁻¹x + tan⁻¹y
tan⁻¹x + tan⁻¹y
tan⁻¹x + tan⁻¹y
tan⁻¹x − tan⁻¹y
2tan⁻¹x
2tan⁻¹x
2tan⁻¹x
sin⁻¹x + sin⁻¹y
cos⁻¹x + cos⁻¹y
Graphs of Inverse Trigonometric Functions
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Find the principal value of sin⁻¹(1/2). | sin⁻¹(1/2) = π/6 (since sin π/6 = 1/2 and π/6 ∈ [-π/2, π/2]). |
| Q2: Find the principal value of cos⁻¹(−1/2). | cos⁻¹(−1/2) = 2π/3 (since cos 2π/3 = −1/2 and 2π/3 ∈ [0, π]). |
| Q3: Evaluate sin⁻¹(1/2) + cos⁻¹(1/2). | sin⁻¹(1/2) + cos⁻¹(1/2) = π/2 (complementary pair property). |
| Q4: Find the value of tan⁻¹(1) + cot⁻¹(1). | tan⁻¹(1) + cot⁻¹(1) = π/2. |
| Q5: Evaluate sin⁻¹(−1/2). | sin⁻¹(−1/2) = −sin⁻¹(1/2) = −π/6. |
| Q6: Find the value of cos⁻¹(−1/2) using the property. | cos⁻¹(−1/2) = π − cos⁻¹(1/2) = π − π/3 = 2π/3. |
| Q7: Evaluate tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3). | tan⁻¹(1) = π/4. tan⁻¹(2) + tan⁻¹(3) = tan⁻¹((2+3)/(1−6)) = tan⁻¹(−1) = −π/4. Sum = 0. |
| Q8: Find the domain of f(x) = sin⁻¹(2x). | Domain: −1 ≤ 2x ≤ 1 → x ∈ [-1/2, 1/2]. |
| Practise these types of questions to become comfortable with applying Inverse Trigonometric Functions concepts in exam scenarios. | |
All ITF Formulas at a Glance
| Formula | What It Means |
|---|---|
| sin⁻¹(−x) = −sin⁻¹x | Odd function |
| cos⁻¹(−x) = π − cos⁻¹x | Even function (shifted) |
| tan⁻¹(−x) = −tan⁻¹x | Odd function |
| sin⁻¹x + cos⁻¹x = π/2 | Complementary pairs |
| tan⁻¹x + cot⁻¹x = π/2 | Complementary pairs |
| sec⁻¹x + cosec⁻¹x = π/2 | Complementary pairs |
| sin⁻¹(1/x) = cosec⁻¹x | Reciprocal arguments |
| cos⁻¹(1/x) = sec⁻¹x | Reciprocal arguments |
| tan⁻¹(1/x) = cot⁻¹x | Reciprocal arguments |
| tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) | Addition formula |
| 2tan⁻¹x = sin⁻¹(2x/(1+x²)) | Double angle (sin) |
| 2tan⁻¹x = cos⁻¹((1−x²)/(1+x²)) | Double angle (cos) |
| Memorise these formulas for Inverse Trigonometric Functions. They are the key to scoring full marks in this chapter. | |
Common Mistakes in ITF
- Confusing domain and range: Remember that the domain of sin⁻¹x is [-1, 1] and the range is [-π/2, π/2]. The domain of tan⁻¹x is R and the range is (-π/2, π/2).
- Forgetting the principal value branch: Always use the principal value branch when evaluating inverse trigonometric functions.
- Misapplying the addition formulas: The formula tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) has conditions on xy. Check the conditions before applying.
- Confusing sin⁻¹x with (sin x)⁻¹: sin⁻¹x is the inverse sine function, not the reciprocal of sine (which is cosec x).
- Not checking the domain of composite functions: When dealing with composite functions like sin⁻¹(2x), always check the domain of the inner function.
- Forgetting the complementary pair properties: sin⁻¹x + cos⁻¹x = π/2 is a very useful property. Memorise it.
Why ITF Matters for JEE and Boards
- Moderate weightage: Inverse Trigonometric Functions appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
- Foundation for calculus: Inverse trigonometric functions are used in integration and differentiation, making them essential for advanced mathematics.
- Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
- Practical relevance: Inverse trigonometric functions are used in physics, engineering, and computer graphics.
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Frequently Asked Questions — Inverse Trigonometric Functions
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