Summary of Vijeta 2025 | Inverse Trigonometric Functions (ITF) One Shot | Maths | Class 12th Boards

Summary of "Vijeta 2025 | Inverse Trigonometric Functions (ITF) One Shot | Maths | Class 12th Boards"

---

Main Ideas and Concepts Covered:

1. Introduction to Inverse Trigonometric Functions (ITF):
   • Importance of ITF in Class 12 Maths and calculus.
   • ITF is a short but crucial chapter with significant applications in calculus.
   • Emphasis on understanding concepts rather than rote learning for board and competitive exams (CBSE, JEE, CU).
2. Basic Trigonometry Refresher:
   • Right-angled triangle definitions: opposite (perpendicular), adjacent (base), hypotenuse.
   • Six Trigonometric Ratios: sin, cos, tan, cot, sec, cosec.
   • Fundamental identities:
     • sin² θ + cos² θ = 1
     • 1 + tan² θ = sec² θ
     • 1 + cot² θ = csc² θ
3. Definition and Concept of Inverse Trigonometric Functions:
   • ITF gives the angle corresponding to a given trigonometric ratio value.
   • Notation: sin⁻¹ x, cos⁻¹ x, tan⁻¹ x, etc.
   • Principal value and general value concepts:
     • Principal value is the unique value of the inverse function within a restricted domain ensuring it is a function (one-to-one).
     • General values correspond to all possible angles satisfying the trigonometric equation.
4. Graphs of Inverse Trigonometric Functions:
   • Graphs of sin⁻¹ x, cos⁻¹ x, tan⁻¹ x, cot⁻¹ x, sec⁻¹ x, csc⁻¹ x explained with domain and range.
   • Domains and ranges are interchanged from the original trigonometric functions.
   • Restrictions on domains to make functions bijective for inverses.
   • Open and closed interval notations explained (e.g., [−π/2, π/2] for sin⁻¹ x).
5. Principal Value Branches:
   • Importance of restricting domain to principal branches for defining inverse functions uniquely.
   • Examples of principal intervals for various ITFs:
     • sin⁻¹ x: [−π/2, π/2]
     • cos⁻¹ x: [0, π]
     • tan⁻¹ x: (−π/2, π/2)
     • cot⁻¹ x: (0, π)
     • sec⁻¹ x, csc⁻¹ x with respective domains.
6. Properties and Formulas of ITF:
   • Negative inside inverse functions:

     sin⁻¹(−x) = −sin⁻¹ x,
     cos⁻¹(−x) = π − cos⁻¹ x,
     tan⁻¹(−x) = −tan⁻¹ x, etc.

   • Addition formulas for inverse tangent:

     tan⁻¹ x + tan⁻¹ y = tan⁻¹ ((x + y) / (1 − xy))   if xy < 1

   • Complementary relations:

     sin⁻¹ x + cos⁻¹ x = π/2,
     tan⁻¹ x + cot⁻¹ x = π/2

7. Conversion Between Different ITFs:
   • Techniques to convert one inverse trigonometric function into another using triangle relations and Pythagoras theorem.
   • For example, converting tan⁻¹ x into sin⁻¹ y or cos⁻¹ z by expressing sides of a right triangle.
   • Use of identities like:

     sin⁻¹ x = tan⁻¹ (x / √(1 − x²)),
     cos⁻¹ x = sin⁻¹ √(1 − x²)

Category

Educational

Video