# 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.