Properties of Inverse Trigonometric Functions

1. Symmetry:

  • Reciprocal Symmetry:
    • sin⁻¹(1/x) = cos⁻¹(x)
    • cos⁻¹(1/x) = sin⁻¹(x)
  • Complementary Symmetry:
    • sin⁻¹(x) + cos⁻¹(x) = π/2
    • tan⁻¹(x) + cot⁻¹(x) = π/2
    • sec⁻¹(x) + csc⁻¹(x) = π/2

2. Range and Domain:

  • sin⁻¹(x):

    • Domain: [-1, 1]
    • Range: [-π/2, π/2]
  • cos⁻¹(x):

    • Domain: [-1, 1]
    • Range: [0, π]
  • tan⁻¹(x):

    • Domain: All real numbers
    • Range: (-π/2, π/2)
  • cot⁻¹(x):

    • Domain: All real numbers except 0
    • Range: (0, π)
  • sec⁻¹(x):

    • Domain: (-∞, -1] U [1, ∞)
    • Range: [0, π] U [π, 2π]
  • csc⁻¹(x):

    • Domain: (-∞, -1] U [1, ∞)
    • Range: [-π/2, 0] U [0, π/2]

3. Periodicity:

  • Inverse trigonometric functions are not periodic in the same way as their corresponding trigonometric functions. They lack a consistent repetition of values.

4. Trigonometric Identities:

  • Complementary Angle Identities:

    • sin⁻¹(x) = cos⁻¹(√(1 - x²))
    • cos⁻¹(x) = sin⁻¹(√(1 - x²))
    • tan⁻¹(x) = cot⁻¹(1/x)
    • cot⁻¹(x) = tan⁻¹(1/x)
    • sec⁻¹(x) = csc⁻¹(1/x)
    • csc⁻¹(x) = sec⁻¹(1/x)
  • Double Angle Identities:

    • sin⁻¹(2x/√(1 + x²)) = 2sin⁻¹(x)
    • cos⁻¹(2x/√(1 + x²)) = 2cos⁻¹(x)
    • tan⁻¹((2x)/(1 - x²)) = 2tan⁻¹(x)

5. Inverse of Composite Functions:

  • (sin⁻¹(x))' = 1/√(1 - x²)
  • (cos⁻¹(x))' = -1/√(1 - x²)
  • (tan⁻¹(x))' = 1/(1 + x²)
  • (cot⁻¹(x))' = -1/(1 + x²)
  • (sec⁻¹(x))' = 1/(|x|√(x² - 1))
  • (csc⁻¹(x))' = -1/(|x|√(x² - 1))

6. Inverse Trigonometric Functions and Equations:

  • Inverse trigonometric functions are commonly used to solve equations involving trigonometric expressions. Care must be taken to consider the appropriate range and domain for solutions.