Lesson 4-1: Inverse Trigonometric & Hyperbolic Functions

Master inverse trigonometric functions (arcsin, arccos, arctan) and hyperbolic functions (sinh, cosh, tanh) with calculus rules, identities, and real-world applications.

Learning Objectives

Recall domains and principal ranges for arcsin, arccos, and arctan.
Differentiate and integrate expressions involving inverse trig and hyperbolic functions.
Use identities to simplify algebra and evaluate limits.
Model real contexts (e.g., catenary) with hyperbolic functions.

Core Formulas

Inverse Trigonometric Derivatives

\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\arctan x = \frac{1}{1+x^2}

Hyperbolic Function Derivatives

\frac{d}{dx}\sinh x = \cosh x

\frac{d}{dx}\cosh x = \sinh x

\frac{d}{dx}\tanh x = \text{sech}^2 x = 1 - \tanh^2 x

Inverse Trigonometric Functions

Definitions and Domains

arcsin x

Domain: $[-1,1]$

Range: $[-\pi/2, \pi/2]$

arccos x

Domain: $[-1,1]$

Range: $[0, \pi]$

arctan x

Domain: $\mathbb{R}$

Range: $(-\pi/2, \pi/2)$

Key Identities

\arcsin x + \arccos x = \frac{\pi}{2} \text{ for } x \in [-1,1]

\arctan x + \arctan\frac{1}{x} = \frac{\pi}{2} \text{ for } x > 0

Hyperbolic Functions

Definitions

\sinh x = \frac{e^x - e^{-x}}{2}

\cosh x = \frac{e^x + e^{-x}}{2}

\tanh x = \frac{\sinh x}{\cosh x}

Fundamental Identity

\cosh^2 x - \sinh^2 x = 1

Worked Examples

Example 1: Derivative of Inverse Function

Find the derivative of $f(x) = x^2 \arcsin x$ .

Solution:

Using product rule:

f'(x) = 2x \arcsin x + x^2 \cdot \frac{1}{\sqrt{1-x^2}}

= 2x \arcsin x + \frac{x^2}{\sqrt{1-x^2}}

Example 2: Catenary Curve

A hanging chain forms a catenary curve described by $y = a \cosh\frac{x}{a}$ where $a$ is a constant.

Find the derivative:

\frac{dy}{dx} = a \sinh\frac{x}{a} \cdot \frac{1}{a} = \sinh\frac{x}{a}

Practice Problems

Problem 1

Find the derivative of $f(x) = \arctan(2x)$ .

f'(x) = \frac{2}{1+(2x)^2} = \frac{2}{1+4x^2}

Problem 2

Evaluate $\int \frac{dx}{\sqrt{1-x^2}}$ .

\int \frac{dx}{\sqrt{1-x^2}} = \arcsin x + C

Key Takeaways

Inverse trigonometric functions have restricted domains and ranges for uniqueness.
Hyperbolic functions are defined using exponential functions and have useful identities.
Both function types have specific derivative and integral rules that should be memorized.
These functions model real-world phenomena like hanging chains and oscillatory motion.

Continue to Lesson 4-2 for first-order differential equations.

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