Class 12 Notes - Inverse Trigonometric Functions

Download PDF Worksheets
Start a Practice Test

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). They are used to find the angle when the value of the trigonometric function is known.

1. Definition

  • If y = sin-1x, then x = sin y and y is in [-π/2, π/2].
  • If y = cos-1x, then x = cos y and y is in [0, π].
  • If y = tan-1x, then x = tan y and y is in (-π/2, π/2).
  • Similarly for cosec-1x, sec-1x, cot-1x with their respective domains and ranges.

2. Principal Value Branches

  • sin-1x: x ∈ [-1, 1], principal value in [-π/2, π/2]
  • cos-1x: x ∈ [-1, 1], principal value in [0, π]
  • tan-1x: x ∈ ℝ, principal value in (-π/2, π/2)
  • cosec-1x: x ∈ ℝ, |x| ≥ 1, principal value in [-π/2, 0) ∪ (0, π/2]
  • sec-1x: x ∈ ℝ, |x| ≥ 1, principal value in [0, π/2) ∪ (π/2, π]
  • cot-1x: x ∈ ℝ, principal value in (0, π)

3. Properties

  • sin-1(-x) = -sin-1x
  • cos-1(-x) = π - cos-1x
  • tan-1(-x) = -tan-1x
  • sin-1x + cos-1x = π/2
  • tan-1x + cot-1x = π/2
  • sec-1x = cos-1(1/x), x ≥ 1 or x ≤ -1
  • cosec-1x = sin-1(1/x), x ≥ 1 or x ≤ -1

4. Important Results

  • tan-1x + tan-1y = tan-1[(x + y)/(1 - xy)], if xy < 1
  • tan-1x - tan-1y = tan-1[(x - y)/(1 + xy)], if xy > -1
  • 2 tan-1x = tan-1[2x/(1 - x2)] (for x ≠ ±1)

5. Graphs

  • The graphs of inverse trigonometric functions are the reflections of the restricted trigonometric functions about the line y = x.

6. Example Problems

  1. Find the value of sin-1(1/2):
    sin-1(1/2) = π/6
  2. Simplify tan-11 + tan-12 + tan-13:
    tan-11 + tan-12 = tan-1[(1+2)/(1-1×2)] = tan-1(-3) = -tan-13
    So, sum is 0.
  3. Find the principal value of cos-1(-1/2):
    cos-1(-1/2) = 2π/3

7. Applications

  • Solving equations involving trigonometric functions
  • Calculating angles in geometry and physics
  • Used in calculus, especially in integration

8. Practice Questions

  1. Find the value of sin-1(-1).
  2. Simplify tan-11 + tan-12.
  3. Find the principal value of sec-1(2).
  4. If sin-1x + cos-1x = π/2, find x.
  5. Evaluate tan-1(1/√3).

Summary

  • Inverse trigonometric functions help find angles from known trigonometric values.
  • They have restricted domains and ranges (principal values).
  • They are important in higher mathematics, calculus, and real-world applications.