Inverse Trigonometric Functions and Mathematical Limits

Flashcards covering the definitions, domains, ranges, mathematical properties, and series simplification methods for Inverse Trigonometric Functions, concluding with fundamental definitions of Limits.

$\sin^{-1}(x)$ (Domain and Range)

The Domain is $[-1,1]$ and the Range (Principal Value) is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

$\cos^{-1}(x)$ (Domain and Range)

The Domain is $[-1,1]$ and the Range (Principal Value) is $[0, \pi]$.

$\tan^{-1}(x)$ (Domain and Range)

The Domain is $\mathbb{R}$ and the Range (Principal Value) is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

$\cot^{-1}(x)$ (Domain and Range)

The Domain is $\mathbb{R}$ and the Range (Principal Value) is $(0, \pi)$, and it is a decreasing function.

$\sec^{-1}(x)$ (Domain and Range)

The Domain is $(-\infty, -1] \cup [1, \infty)$ and the Range (Principal Value) is $[0, \pi] - \{\frac{\pi}{2}\}$.

$\csc^{-1}(x)$ (Domain and Range)

The Domain is $(-\infty, -1] \cup [1, \infty)$ and the Range (Principal Value) is $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$.

Increasing Functions in ITF

$\sin^{-1}(x)$ and $\tan^{-1}(x)$ are classified as increasing functions.

Decreasing Functions in ITF

$\cos^{-1}(x)$ and $\cot^{-1}(x)$ are classified as decreasing functions.

Odd Functions (P-1)

$\sin^{-1}(-x) = -\sin^{-1}(x)$, $\tan^{-1}(-x) = -\tan^{-1}(x)$, and $\csc^{-1}(-x) = -\csc^{-1}(x)$.

Neno Functions

Functions that are 'Neither Even Nor Odd', referring to $\cos^{-1}(x)$, $\cot^{-1}(x)$, and $\sec^{-1}(x)$ where, for example, $\cos^{-1}(-x) = \pi - \cos^{-1}(x)$.

Property 2 (P-2)

Complementary angle identities where $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$, $\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}$, and $\sec^{-1}(x) + \csc^{-1}(x) = \frac{\pi}{2}$.

Property 4 (P-4)

Reciprocal relationships where $\csc^{-1}(x) = \sin^{-1}(\frac{1}{x})$ and $\sec^{-1}(x) = \cos^{-1}(\frac{1}{x})$ for $x \in (-\infty, -1] \cup [1, \infty)$, and $\cot^{-1}(x) = \tan^{-1}(\frac{1}{x})$ for $x > 0$.

Addition of Angles for Inverse Tangent

$\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}(\frac{x+y}{1-xy})$ provided that $x,y > 0$ and $xy < 1$. If $x,y > 0$ and $xy > 1$, the result is $\pi + \tan^{-1}(\frac{x+y}{1-xy})$.

Telescopic Series for tan^-1

A technique using the formula $\tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}(\frac{a-b}{1+ab})$ to convert a series into a difference of two quantities for summation simplification.

Limit

The value that a function $y = f(x)$ approaches when $x$ tends to a value $a$, denoted as $\lim_{x \rightarrow a} f(x)$.

LHL (Left Hand Limit)

The limit value approached from the left side of a point $a$, given by $\lim_{x \rightarrow a^-} f(x)$ or $\lim_{h \rightarrow 0} f(a-h)$, where $h$ is a very small positive quantity.

RHL (Right Hand Limit)

The limit value approached from the right side of a point $a$, given by $\lim_{x \rightarrow a^+} f(x)$ or $\lim_{h \rightarrow 0} f(a+h)$, where $h$ is a very small positive quantity.