Inverse Trigonometric Functions - Graphs of inverse trigonometric functions

Definition and Bijectivity: Trigonometric functions are periodic and hence not one-to-one over their entire domain. To define inverse trigonometric functions, their domains are restricted to specific intervals (Principal Value Branches) where the functions are bijective. Visually, the graph of an inverse function is the reflection of the original function's graph across the line $y = x$.

Graph of $y = \sin^{-1} x$: The domain is $[-1, 1]$ and the principal value range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Visually, it is an 'S-shaped' curve passing through the origin $(0, 0)$. It is strictly increasing, starting from the point $(-1, -\frac{\pi}{2})$ and ending at $(1, \frac{\pi}{2})$.

Graph of $y = \cos^{-1} x$: The domain is $[-1, 1]$ and the principal value range is $[0, \pi]$. Visually, this is a strictly decreasing curve that starts at $(-1, \pi)$, crosses the y-axis at $(0, \frac{\pi}{2})$, and ends at the point $(1, 0)$. Unlike the sine inverse, it lies entirely above or on the x-axis.

Graph of $y = \tan^{-1} x$: The domain is the set of all real numbers $\mathbb{R}$, and the principal value range is the open interval $(-\frac{\pi}{2}, \frac{\pi}{2})$. Visually, the graph features two horizontal asymptotes at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$. The curve increases from left to right, passing through the origin and flattening out as it approaches the asymptotes.

Graph of $y = \text{cosec}^{-1} x$ and $y = \sec^{-1} x$: These functions have a domain of $(-\infty, -1] \cup [1, \infty)$. Visually, there is a visible 'gap' in the graph between $x = -1$ and $x = 1$. The graph of $\text{cosec}^{-1} x$ has a horizontal asymptote at $y = 0$, while $\sec^{-1} x$ has one at $y = \frac{\pi}{2}$.

Graph of $y = \cot^{-1} x$: The domain is $\mathbb{R}$ and the range is $(0, \pi)$. Visually, it is a strictly decreasing curve. It approaches the horizontal asymptote $y = \pi$ as $x \to -\infty$ and approaches $y = 0$ as $x \to \infty$.

Symmetry and Odd/Even properties: The graphs of $\sin^{-1} x$, $\tan^{-1} x$, and $\text{cosec}^{-1} x$ are symmetric with respect to the origin (odd functions), meaning $\sin^{-1}(-x) = -\sin^{-1}(x)$. However, $\cos^{-1} x$, $\sec^{-1} x$, and $\cot^{-1} x$ do not show origin symmetry; instead, their values follow the relation $f(-x) = \pi - f(x)$ due to their shifted ranges.