Inverse Trigonometric Functions

Definition and Properties of Inverse Functions

Inverse functions are fundamental in calculus, allowing us to reverse the effect of a function. If f is a one-to-one function defined from domain D to range R, then its inverse, denoted f-1, satisfies:

• f-1(y) = x if and only if y = f(x)

• f(f-1(y)) = y and f-1(f(x)) = x for all x ∈ D and y ∈ R

A function that is strictly monotonic (either strictly increasing or strictly decreasing) on an interval has an inverse on that interval.

Example:

• The function f(x) = x^2 is strictly monotonic on [0, ∞), so its inverse is f-1(y) = √y for y ∈ [0, ∞).

Inverse of the Sine Function

Definition and Domain/Range

The sine function is strictly monotonic on the interval [−π/2, π/2] and its range is [−1, 1]. Its inverse is denoted by sin-1 or arcsin, defined from [−1, 1] to [−π/2, π/2]:

• , ,

Key Properties:

• for all such that

• for all such that

Examples:

• , since

• , since and

Derivative of the Inverse Sine Function

The derivative of the inverse sine function is given by:

• For a composite function:

Examples:

• Find the derivative of

• Find the derivative of

Inverse of the Cosine Function

Definition and Domain/Range

The cosine function is strictly monotonic on [0, π] and its range is [−1, 1]. Its inverse is denoted by cos-1 or arccos, defined from [−1, 1] to [0, π]:

• , ,

Key Properties:

• for all such that

• for all such that

Examples:

• , since

• , since and

Derivative of the Inverse Cosine Function

The derivative of the inverse cosine function is:

• For a composite function:

Examples:

• Find the derivative of

• Find the derivative of

Summary Table: Inverse Trigonometric Functions and Their Derivatives

Additional info: These notes cover the definitions, domains, ranges, and derivatives of the inverse sine and cosine functions, with examples and key properties. For a complete study, students should also review the inverse tangent, cotangent, secant, and cosecant functions, which follow similar principles.