Exhaustive Guide to Inverse Trigonometric Functions

Philosophies and Foundations of Inverse Trigonometric Functions

• Felix Klein Quote: "Mathematics, in general, is fundamentally the science of self-evident things."
• Pre-requisites for Inversibility: A function $f$ has an inverse, denoted as $f^{-1}$, if and only if $f$ is one-one (injective) and onto (surjective).
• Trigonometric Limitations: Trigonometric functions are periodic and not one-one or onto over their natural domains ($R$). Therefore, they do not naturally possess inverses over their entire domains.
• Necessary Restrictions: To define inverse functions, the domains and ranges of trigonometric functions must be restricted to specific intervals where the function becomes bijective (both one-one and onto).
• Importance in Calculus: Inverse trigonometric functions are essential for defining many integrals.
• Applications: These concepts are widely applied in various fields of science and engineering.

Fundamental Concepts of Bijection and Inversion

• Definition of Inverse: If $f : X \rightarrow Y$ such that $f(x) = y$ is one-one and onto, then there exists a unique function $g : Y \rightarrow X$ such that $g(y) = x$.
• Relational Properties:
  • $Domain(g) = Range(f)$
  • $Range(g) = Domain(f)$
  • The function $g$ is denoted as $f^{-1}$.
• Inversion of the Inverse: The function $g$ is also one-one and onto, and the inverse of $g$ is $f$. Thus, $(f^{-1})^{-1} = f$.
• Composition Identities:
  • $(f^{-1} \circ f)(x) = f^{-1}(f(x)) = x$ for all $x \in X$
  • $(f \circ f^{-1})(y) = f(f^{-1}(y)) = y$ for all $y \in Y$

Detailed Analysis of the Inverse Sine Function ($sin^{-1}x$)

• Natural Sine Function: $sine : R \rightarrow [-1, 1]$.
• Injectivity Restriction: By restricting the domain to $[-\frac{\pi}{2}, \frac{\pi}{2}]$, the sine function becomes one-one and onto with a range of $[-1, 1]$.
• Alternate Restricted Domains: The sine function is also bijective over intervals such as $[-\frac{3\pi}{2}, -\frac{\pi}{2}]$ or $[\frac{\pi}{2}, \frac{3\pi}{2}]$.
• Definition of $sin^{-1}$: A function with domain $[-1, 1]$ and range restricted to one of the bijective branches.
• Principal Value Branch: The specific branch with the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ is designated as the principal value branch for $sin^{-1}$.
• Operational Identity Constraints:
  • $sin(sin^{-1} x) = x$ if $-1 \leq x \leq 1$
  • $sin^{-1}(sin x) = x$ if $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$
• Graphical Construction: The graph of $y = sin^{-1} x$ is a reflection of $y = sin x$ across the line $y = x$. This is equivalent to interchanging the $x$ and $y$ axes.

Detailed Analysis of Inverse Trigonometric Functions

• Inverse Cosine ($cos^{-1}x$):
  • Domain: $[-1, 1]$
  • Principal Value Branch (Range): $[0, \pi]$
  • Natural Co-domain: Bijective over intervals like $[-\pi, 0]$, $[0, \pi]$, and $[\pi, 2\pi]$.
• Inverse Cosecant ($cosec^{-1}x$):
  • Original Function: $cosec(x) = \frac{1}{sin(x)}$, where domain is $R - \{x : x = n\pi, n \in Z\}$.
  • Domain of Inverse: $R - (-1, 1)$, which means $y \geq 1$ or $y \leq -1$.
  • Principal Value Branch: $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$.
• Inverse Secant ($sec^{-1}x$):
  • Original Function: $sec(x) = \frac{1}{cos(x)}$, where domain is $R - \{x : x = (2n + 1)\frac{\pi}{2}, n \in Z\}$.
  • Domain of Inverse: $R - (-1, 1)$.
  • Principal Value Branch: $[0, \pi] - \{\frac{\pi}{2}\}$.
• Inverse Tangent ($tan^{-1}x$):
  • Domain: $R$ (all real numbers).
  • Principal Value Branch: $(-\frac{\pi}{2}, \frac{\pi}{2})$ (open interval, as $\\tan$ is undefined at odd multiples of $\frac{\pi}{2}$).
• Inverse Cotangent ($cot^{-1}x$):
  • Domain: $R$
  • Principal Value Branch: $(0, \pi)$.

Summary Table of Principal Branches

• Inverse Sine: Domain: $[-1, 1]$, Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$
• Inverse Cosine: Domain: $[-1, 1]$, Range: $[0, \pi]$
• Inverse Cosecant: Domain: $R - (-1, 1)$, Range: $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$
• Inverse Secant: Domain: $R - (-1, 1)$, Range: $[0, \pi] - \{\frac{\pi}{2}\}$
• Inverse Tangent: Domain: $R$, Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$
• Inverse Cotangent: Domain: $R$, Range: $(0, \pi)$

Critical Remarks and Definitions

• Notation Warning: $sin^{-1}x$ is a notation for the inverse function and is NOT the same as $(sin x)^{-1}$. Specifically, $(sin x)^{-1} = \frac{1}{sin x}$.
• Principal Value: The value of an inverse trigonometric function lying within the range of its principal value branch is called the "Principal Value".
• Default Branch: If no specific branch is mentioned in a problem, it is assumed to be the principal value branch.

Practical Examples and Solutions

• Example 1: Find principal value of $sin^{-1}(\frac{1}{\sqrt{2}})$.
  • Let $y = sin^{-1}(\frac{1}{\sqrt{2}})$.
  • Then $sin(y) = \frac{1}{\sqrt{2}}$.
  • Since $sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ and $\frac{\pi}{4} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, the principal value is $\frac{\pi}{4}$.
• Example 2: Find principal value of $cot^{-1}(-\frac{1}{\sqrt{3}})$.
  • Let $y = cot^{-1}(-\frac{1}{\sqrt{3}})$.
  • Then $cot(y) = -\frac{1}{\sqrt{3}}$.
  • $cot(y) = -cot(\frac{\pi}{3}) = cot(\pi - \frac{\pi}{3}) = cot(\frac{2\pi}{3})$.
  • Since $\frac{2\pi}{3} \in (0, \pi)$, the principal value is $\frac{2\pi}{3}$.
• Miscellaneous Example: Evaluate $sin^{-1}(sin(\frac{3\pi}{5}))$.
  • $sin^{-1}(sin(x)) = x$ only if $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.
  • $\frac{3\pi}{5}$ is NOT in the principal range because $\frac{3\pi}{5} > \frac{\pi}{2}$.
  • Rewrite $sin(\frac{3\pi}{5})$ as $sin(\pi - \frac{3\pi}{5}) = sin(\frac{2\pi}{5})$.
  • Since $\frac{2\pi}{5} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, the value is $\frac{2\pi}{5}$.

Fundamental Identities and Algebraic Properties

• Substitution Methods:
  • To simplify $sin^{-1}(2x\sqrt{1-x^2})$, setting $x = sin(\theta)$ or $x = cos(\theta)$ is effective.
  • Result (i): $sin^{-1}(2x\sqrt{1-x^2}) = 2sin^{-1}x$ for $-\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}$.
  • Result (ii): $sin^{-1}(2x\sqrt{1-x^2}) = 2cos^{-1}x$ for $\frac{1}{\sqrt{2}} \leq x \leq 1$.
• Specific Trig Simplifications:
  • Simplifying $tan^{-1}(\frac{cos(x)}{1 - sin(x)})$: Use the identity $cos(x) = cos^2(\frac{x}{2}) - sin^2(\frac{x}{2})$ and $1 - sin(x) = (cos(\frac{x}{2}) - sin(\frac{x}{2}))^2$.
  • This leads to $tan^{-1}(tan(\frac{\pi}{4} + \frac{x}{2})) = \frac{\pi}{4} + \frac{x}{2}$.
• Simplest Form of $cot^{-1}(\frac{1}{\sqrt{x^2 - 1}})$ for $x > 1$:
  • Put $x = sec(\theta)$.
  • $cot^{-1}(\frac{1}{\sqrt{sec^2(\theta) - 1}}) = cot^{-1}(\frac{1}{tan(\theta)}) = cot^{-1}(cot(\theta)) = \theta = sec^{-1}x$.

Exercise Reference Concepts

• Inverse sine of triple angle: $3sin^{-1}x = sin^{-1}(3x - 4x^3)$.
• Inverse cosine of triple angle: $3cos^{-1}x = cos^{-1}(4x^3 - 3x)$.
• Complex Tan Substitution: For expressions like $tan^{-1}(\frac{3a^2x - x^3}{a^3 - 3ax^2})$, putting $x = a\,tan(\theta)$ is appropriate.

Historical Context of Trigonometry

• Indian Origins: The study of trigonometry originated in India. Key contributors include:
  • Aryabhata ($476\,A.D.$).
  • Brahmagupta ($598\,A.D.$).
  • Bhaskara I ($600\,A.D.$), who provided formulas for sine of angles greater than $90^{\circ}$.
  • Bhaskara II ($1114\,A.D.$), who gave exact expressions for sines and cosines of $18^{\circ}$, $36^{\circ}$, $54^{\circ}$, and $72^{\circ}$.
• Transmission of Knowledge: Indian results traveled to Arabia and then to Europe. The Indian approach was adopted globally due to its clarity compared to Greek methods.
• Modern Terminology:
  • The concept of "sine" is a contribution of the Sanskrit Siddhantas.
  • Yuktibhasa (16th Century Malayalam work) contains a proof for the expansion of $sin(A + B)$.
  • Notation Suggestion: The symbols $sin^{-1}x$ and $cos^{-1}x$ for arc sine and arc cosine were suggested by Sir John F.W. Hersehel in $1813$.
• Thales of Miletus ($600\,B.C.$): Associated with height and distance problems. He determined the height of the Great Pyramid of Egypt by comparing shadow ratios: $\frac{H}{S} = \frac{h}{s} = tan(\text{sun's altitude})$.