Inverse Trigonometry

• Inverse trigonometry deals with inverse trigonometric functions.
• These functions are the inverses of standard trigonometric functions, including sine, cosine, tangent, cosecant, secant, and cotangent.
• Inverse trigonometric functions find an angle from a known trigonometric ratio value.

Basic Inverse Trigonometric Functions

• Arcsine (sin⁻¹x or arcsin x): It is the inverse of the sine function.
• Arccosine (cos⁻¹x or arccos x): It is the inverse of the cosine function.
• Arctangent (tan⁻¹x or arctan x): It is the inverse of the tangent function.
• Arccosecant (cosec⁻¹x or arccsc x): It is the inverse of the cosecant function.
• Arcsecant (sec⁻¹x or arcsec x): It is the inverse of the secant function.
• Arccotangent (cot⁻¹x or arccot x): It is the inverse of the cotangent function.

Domains and Ranges

• Understanding domains and ranges is crucial for inverse trigonometric functions.
• Principal Value Branches: Standard intervals are chosen to make inverse functions single-valued.

Function: sin⁻¹x

• Domain: [-1, 1]
• Range (Principal Value Branch): [-π/2, π/2]
• sin⁻¹x provides the angle whose sine is x.

Function: cos⁻¹x

• Domain: [-1, 1]
• Range (Principal Value Branch): [0, π]
• cos⁻¹x provides the angle whose cosine is x.

Function: tan⁻¹x

• Domain: (-∞, ∞)
• Range (Principal Value Branch): (-π/2, π/2)
• tan⁻¹x provides the angle whose tangent is x.

Function: cosec⁻¹x

• Domain: (-∞, -1] ∪ [1, ∞)
• Range (Principal Value Branch): [-π/2, π/2] - {0}
• cosec⁻¹x provides the angle whose cosecant is x.

Function: sec⁻¹x

• Domain: (-∞, -1] ∪ [1, ∞)
• Range (Principal Value Branch): [0, π] - {π/2}
• sec⁻¹x provides the angle whose secant is x.

Function: cot⁻¹x

• Domain: (-∞, ∞)
• Range (Principal Value Branch): (0, π)
• cot⁻¹x provides the angle whose cotangent is x.

Properties of Inverse Trigonometric Functions

• sin⁻¹(sin x) = x, for x ∈ [-π/2, π/2]
• cos⁻¹(cos x) = x, for x ∈ [0, π]
• tan⁻¹(tan x) = x, for x ∈ (-π/2, π/2)
• cosec⁻¹(cosec x) = x, for x ∈ [-π/2, π/2] - {0}
• sec⁻¹(sec x) = x, for x ∈ [0, π] - {π/2}
• cot⁻¹(cot x) = x, for x ∈ (0, π)

Inverse Properties

• sin(sin⁻¹x) = x, for x ∈ [-1, 1]
• cos(cos⁻¹x) = x, for x ∈ [-1, 1]
• tan(tan⁻¹x) = x, for x ∈ (-∞, ∞)
• cosec(cosec⁻¹x) = x, for x ∈ (-∞, -1] ∪ [1, ∞)
• sec(sec⁻¹x) = x, for x ∈ (-∞, -1] ∪ [1, ∞)
• cot(cot⁻¹x) = x, for x ∈ (-∞, ∞)

Negative Argument Properties

• sin⁻¹(-x) = -sin⁻¹x
• tan⁻¹(-x) = -tan⁻¹x
• cosec⁻¹(-x) = -cosec⁻¹x
• cos⁻¹(-x) = π - cos⁻¹x
• sec⁻¹(-x) = π - sec⁻¹x
• cot⁻¹(-x) = π - cot⁻¹x

Reciprocal Identities

• cosec⁻¹x = sin⁻¹(1/x)
• sec⁻¹x = cos⁻¹(1/x)
• cot⁻¹x = tan⁻¹(1/x), if x > 0
• cot⁻¹x = π + tan⁻¹(1/x), if x < 0

Sum and Difference Identities

• 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
• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• sec⁻¹x + cosec⁻¹x = π/2

Double Angle Formulas

• 2tan⁻¹x = sin⁻¹((2x)/(1+x²)), |x| ≤ 1
• 2tan⁻¹x = cos⁻¹((1-x²)/(1+x²)), x ≥ 0
• 2tan⁻¹x = tan⁻¹((2x)/(1-x²)), -1 < x < 1

Examples and Applications

• Simplifying expressions: Properties can simplify expressions involving inverse trigonometric functions.
• Solving equations: Properties and identities are used in solving equations that have inverse trigonometric functions.
• Applications in Physics and Engineering: Finding angles in navigation, optics, and mechanics involves inverse trigonometric functions.

Tips for Solving Problems

• Learn the domains and ranges of the inverse trigonometric functions.
• Familiarize yourself with the properties and identities.
• Simplify expressions and practice solving equations.
• Account for the principal value branch when finding values of inverse trigonometric functions.
• Simplify expressions using suitable substitutions.

Differentiation of Inverse Trigonometric Functions

• d/dx (sin⁻¹x) = 1/√(1-x²)
• d/dx (cos⁻¹x) = -1/√(1-x²)
• d/dx (tan⁻¹x) = 1/(1+x²)
• d/dx (cosec⁻¹x) = -1/(x√(x²-1))
• d/dx (sec⁻¹x) = 1/(x√(x²-1))
• d/dx (cot⁻¹x) = -1/(1+x²)

Integration of Inverse Trigonometric Functions

• Integration by parts can solve integrals of inverse trigonometric functions.
• ∫ sin⁻¹x dx = x sin⁻¹x + √(1-x²) + C
• ∫ cos⁻¹x dx = x cos⁻¹x - √(1-x²) + C
• ∫ tan⁻¹x dx = x tan⁻¹x - (1/2)ln(1+x²) + C

Common Mistakes to Avoid

• Forgetting domain and range restrictions.
• Applying properties and identities incorrectly.
• Not considering the principal value branch.
• Making algebraic errors during simplification.

Advanced Topics

• Complex arguments: Consideration of inverse trigonometric functions with complex arguments.
• Hyperbolic functions: Examining the relationship between inverse trigonometric and inverse hyperbolic functions.
• Applications in complex analysis: Use in solving complex integrals and series.