Inverse Trigonometric Functions and Their Derivatives: Study Notes

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]:

• $ext{sin}^{-1}(y) = x$, $-1 ext{≤} y ext{≤} 1$ $ext{⇔}$ $y = ext{sin}x$, $- rac{ ext{π}}{2} ext{≤} x ext{≤} rac{ ext{π}}{2}$

Key Properties:

• $ext{sin}^{-1}( ext{sin}x) = x$ for all $x$ such that $- rac{ ext{π}}{2} ext{≤} x ext{≤} rac{ ext{π}}{2}$

• $ext{sin}( ext{sin}^{-1}y) = y$ for all $y$ such that $-1 ext{≤} y ext{≤} 1$

Examples:

• $ext{sin}^{-1}igg( rac{2}{3}igg) = x$, since $rac{2}{3} ext{∈} [−1, 1]$

• $ext{sin}^{-1}igg( rac{1}{2}igg) = rac{ ext{π}}{6}$, since $rac{1}{2} = ext{sin}igg( rac{ ext{π}}{6}igg)$ and $rac{ ext{π}}{6} ext{∈} [− rac{ ext{π}}{2}, rac{ ext{π}}{2}]$

Derivative of the Inverse Sine Function

The derivative of the inverse sine function is given by:

• $rac{d}{dx}[ ext{sin}^{-1}(x)] = rac{1}{ ext{√}(1-x^2)}$

• For a composite function: $rac{d}{dx}[ ext{sin}^{-1}(g(x))] = rac{1}{ ext{√}(1-[g(x)]^2)} imes g'(x)$

Examples:

• Find the derivative of $f(x) = x^7 ext{sin}^{-1}(x)$

• Find the derivative of $f(x) = ext{sin}^{-1}igg( rac{1}{x}igg)$

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]:

• $ext{cos}^{-1}(y) = x$, $-1 ext{≤} y ext{≤} 1$ $ext{⇔}$ $y = ext{cos}x$, $0 ext{≤} x ext{≤} ext{π}$

Key Properties:

• $ext{cos}^{-1}( ext{cos}x) = x$ for all $x$ such that $0 ext{≤} x ext{≤} ext{π}$

• $ext{cos}( ext{cos}^{-1}y) = y$ for all $y$ such that $-1 ext{≤} y ext{≤} 1$

Examples:

• $ext{cos}^{-1}igg( rac{3}{5}igg) = x$, since $rac{3}{5} ext{∈} [−1, 1]$

• $ext{cos}^{-1}igg( rac{1}{2}igg) = rac{ ext{π}}{3}$, since $rac{1}{2} = ext{cos}igg( rac{ ext{π}}{3}igg)$ and $rac{ ext{π}}{3} ext{∈} [0, ext{π}]$

Derivative of the Inverse Cosine Function

The derivative of the inverse cosine function is:

• $rac{d}{dx}[ ext{cos}^{-1}(x)] = - rac{1}{ ext{√}(1-x^2)}$

• For a composite function: $rac{d}{dx}[ ext{cos}^{-1}(g(x))] = - rac{1}{ ext{√}(1-[g(x)]^2)} imes g'(x)$

Examples:

• Find the derivative of $f(x) = rac{ ext{cos}(x)}{x^3}$

• Find the derivative of $f(x) = ext{cos}^{-1}( ext{√}x)$

Summary Table: Inverse Trigonometric Functions and Their Derivatives

Function	Domain	Range	Inverse Notation	Derivative
Sine	$[-1, 1]$	$[- rac{ ext{π}}{2}, rac{ ext{π}}{2}]$	$ext{sin}^{-1}(x)$ or $ext{arcsin}(x)$	$rac{1}{ ext{√}(1-x^2)}$
Cosine	$[-1, 1]$	$[0, ext{π}]$	$ext{cos}^{-1}(x)$ or $ext{arccos}(x)$	$- rac{1}{ ext{√}(1-x^2)}$

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.