Inverse Trigonometric Functions

For a function to have an inverse, it must be one-to-one (bijective). Since trigonometric functions are periodic and not one-to-one over their entire domains, we restrict their domains to create principal functions that are invertible.

The inverse trigonometric functions are the inverse functions of the trigonometric functions with restricted domains. They are used to obtain an angle from any of the angle’s trigonometric ratios.

Graphical Approach: A function f is invertible if and only if no horizontal line intersects its graph more than once (Horizontal Line Test).

The restricted function sin: [-π/2, π/2] → [-1, 1] is called the principal sine function.

Why this restriction? The sine function is one-to-one on [-π/2, π/2] and covers the full range of [-1, 1].

Visualization: On the unit circle, this corresponds to the right half of the circle (quadrants I and IV).

The restricted function cos: [0, π] → [-1, 1] is called the principal cosine function.

Why this restriction? The cosine function is one-to-one on [0, π] and covers the full range of [-1, 1].

Visualization: On the unit circle, this corresponds to the upper half of the circle (quadrants I and II).

The restricted function tan: (-π/2, π/2) → (-∞, ∞) is called the principal tangent function.

Note: The endpoints are not included because tan(±π/2) is undefined.

Visualization: This corresponds to the right half of the unit circle excluding the vertical asymptotes.

Principal Cotangent: cot: (0, π) → (-∞, ∞)

Principal Secant: sec: [0, π] \ {π/2} → (-∞, -1] ∪ [1, ∞)

Principal Cosecant: cosec: [-π/2, π/2] \ {0} → (-∞, -1] ∪ [1, ∞)

Each restriction is chosen to make the function one-to-one while covering its entire range.

Definition: sin⁻¹: [-1, 1] → [-π/2, π/2] such that y = sin⁻¹(x) if and only if x = sin(y) for y ∈ [-π/2, π/2]

Domain: [-1, 1] (all possible sine values)

Range: [-π/2, π/2] (principal values)

Key Property: sin(sin⁻¹(x)) = x for x ∈ [-1, 1] and sin⁻¹(sin(y)) = y for y ∈ [-π/2, π/2]

Definition: cos⁻¹: [-1, 1] → [0, π] such that y = cos⁻¹(x) if and only if x = cos(y) for y ∈ [0, π]

Domain: [-1, 1] (all possible cosine values)

Range: [0, π] (principal values)

Key Property: cos(cos⁻¹(x)) = x for x ∈ [-1, 1] and cos⁻¹(cos(y)) = y for y ∈ [0, π]

Definition: tan⁻¹: (-∞, ∞) → (-π/2, π/2) such that y = tan⁻¹(x) if and only if x = tan(y) for y ∈ (-π/2, π/2)

Domain: All real numbers

Range: (-π/2, π/2) (open interval, excludes endpoints)

Key Property: tan(tan⁻¹(x)) = x for all x ∈ ℝ and tan⁻¹(tan(y)) = y for y ∈ (-π/2, π/2)

1. Reciprocal Relations:

• sin⁻¹(1/x) = cosec⁻¹(x), for x ≥ 1 or x ≤ -1
• cos⁻¹(1/x) = sec⁻¹(x), for x ≥ 1 or x ≤ -1
• tan⁻¹(1/x) = cot⁻¹(x), for x > 0

2. Complementary Relations:

• sin⁻¹(x) + cos⁻¹(x) = π/2, for x ∈ [-1, 1]
• tan⁻¹(x) + cot⁻¹(x) = π/2, for x ∈ ℝ
• sec⁻¹(x) + cosec⁻¹(x) = π/2, for x ≥ 1

3. Negative Argument Relations:

• sin⁻¹(-x) = -sin⁻¹(x), for x ∈ [-1, 1]
• cos⁻¹(-x) = π – cos⁻¹(x), for x ∈ [-1, 1]
• tan⁻¹(-x) = -tan⁻¹(x), for x ∈ ℝ

Key Results:

• The functions sin⁻¹(x), cos⁻¹(x), tan⁻¹(x), etc. are called inverse trigonometric functions.
• sin⁻¹(x) is a number (angle) whose sine is x.

Important Caution:

• (sin x)⁻¹ = 1/sin x = cosec x ≠ sin⁻¹(x)
• sin⁻¹(x) means the inverse sine function, not the reciprocal of sine
• This distinction applies to all inverse trigonometric functions

Special Results:

• sin⁻¹(x) + sin⁻¹(y) = sin⁻¹(x√(1-y²) + y√(1-x²)), if x² + y² ≤ 1
• sin⁻¹(x) – sin⁻¹(y) = sin⁻¹(x√(1-y²) – y√(1-x²)), if x² + y² ≤ 1
• Similar formulas exist for cos⁻¹ and tan⁻¹