Unacademy JEE: Mathematics Session on Inverse Trigonometric Functions

Introduction

• Welcome to the Unacademy JEE channel, Mathematics session.
• Ensure regular attendance for all sessions for optimal learning.
• Previous sessions by Jayant sir and Paras sir for Physics and Chemistry.
• Today’s topic: Inverse Trigonometric Functions (ITF).
• Part of the JEE Live Daily series, providing regular lectures.
• Tips for effective learning:
  • Take notes diligently.
  • Apply concepts by solving questions.

About the Instructor

• Name: Samir Chincholikar
• Graduate of IIT Roorkee
• Experience in teaching for JEE preparations.

Inverse Trigonometric Functions (ITF)

Concept Recap

• Invertible Functions Must be Bijective:
  • A function must be 1-1 (injective) and onto (surjective) to have an inverse.
• Domain and Codomain of Inverse Functions:
  • Domain and codomain of the original function swap places in its inverse.
• Graphical Understanding:
  • Graphs of f and f⁻¹ are mirror images along the line y = x.
  • Image of point (x₁, y₁) in y = x is (y₁, x₁).
• Asymptotes:
  • Vertical asymptotes of f become horizontal asymptotes in f⁻¹ and vice versa.

Sine Inverse (sin⁻¹ x)

• Original Function: y = sin x
  • Issues: sin x is not one-to-one or onto over its entire domain.
• Restricting Domain:
  • Restrict domain to [−π/2, π/2] to make sin x bijective.
• Inverse Function:
  • sin⁻¹ x maps from [−1, 1] to [−π/2, π/2].
  • Graph is reflection of y = sin x on y = x.

Cosine Inverse (cos⁻¹ x)

• Original Function: y = cos x
  • Issues: cos x is not one-to-one or onto over its entire domain.
• Restricting Domain:
  • Restrict domain to [0, π] to make cos x bijective.
• Inverse Function:
  • cos⁻¹ x maps from [−1, 1] to [0, π].
  • Graph is reflection of y = cos x on y = x.

Tangent Inverse (tan⁻¹ x)

• Original Function: y = tan x
  • Issues: tan x is not one-to-one over its entire domain.
• Restricting Domain:
  • Restrict domain to (−π/2, π/2) to make tan x bijective.
• Inverse Function:
  • tan⁻¹ x maps from ℝ (all real numbers) to (−π/2, π/2).
  • Graph includes horizontal asymptotes at y = π/2 and y = −π/2.

Cotangent Inverse (cot⁻¹ x)

• Original Function: y = cot x
  • Issues: cot x is not one-to-one over its entire domain.
• Restricting Domain:
  • Restrict domain to (0, π) to make cot x bijective.
• Inverse Function:
  • cot⁻¹ x maps from ℝ to (0, π).
  • Graph is reflection of y = cot x on y = x.

Secant Inverse (sec⁻¹ x)

• Original Function: y = sec x
  • Issues: sec x is not one-to-one over its entire domain.
• Restricting Domain:
  • Use two disjoint intervals: [0, π/2) ∪ (π/2, π).
• Inverse Function:
  • sec⁻¹ x maps from (−∞, −1] ∪ [1, ∞) to [0, π/2) ∪ (π/2, π].
  • Graph reflects vertical asymptotes to horizontal asymptotes.

Cosecant Inverse (csc⁻¹ x)

• Original Function: y = csc x
  • Issues: csc x is not one-to-one over its entire domain.
• Restricting Domain:
  • Use two disjoint intervals: [−π/2, 0) ∪ (0, π/2].
• Inverse Function:
  • csc⁻¹ x maps from (−∞, −1] ∪ [1, ∞) to [−π/2, 0) ∪ (0, π/2].
  • Graph reflects vertical asymptotes to horizontal asymptotes.

Example Problems

Problem 1

• Question: Find the value of tan(cos⁻¹(− √3 /2) + π/3)
• Solution Outline:
  • Identify angle for cos⁻¹(− √3 /2) in [0, π].
  • Solve tan for the resulting expression.
  • Answer: B

Problem 2

• Question: Find the domain of fx = √(π/4 - sin⁻¹ x).
• Solution Outline:
  • Sin inverse x must be in [−1,1] due to its range.
  • Set inequality π/4 - sin⁻¹ x ≥ 0.
  • Determine x range by solving the inequality graphically.
  • Answer: C

Homework Problems

1. Problem: Determine the domain of f(x) = log⁻¹(10/(x⁴ + 64x))
2. Problem: JEE Advance 2015: Find all (a, b) value for which a given ITF equation holds.