Geometry and Trigonometry - Inverse Trigonometric Functions

Domain Restriction for Invertibility: To define inverse functions, the domain of the original trigonometric functions must be restricted so they are one-to-one (bijective). For $\sin(x)$, we restrict the domain to $[-\frac{\pi}{2}, \frac{\pi}{2}]$; for $\cos(x)$, to $[0, \pi]$; and for $\tan(x)$, to $(-\frac{\pi}{2}, \frac{\pi}{2})$. Visually, this selects a single monotonic 'branch' of the periodic wave that passes the horizontal line test.

Inverse Sine (Arcsin): Defined as $y = \arcsin(x)$ or $y = \sin^{-1}(x)$ where the domain is $x \in [-1, 1]$ and the range is $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$. The graph is a reflection of the $\sin(x)$ curve over the line $y=x$, appearing as an S-shaped curve passing through the origin $(0,0)$ and terminating at endpoints $(-1, -\frac{\pi}{2})$ and $(1, \frac{\pi}{2})$.

Inverse Cosine (Arccos): Defined as $y = \arccos(x)$ or $y = \cos^{-1}(x)$ where the domain is $x \in [-1, 1]$ and the range is $y \in [0, \pi]$. Visually, the graph is strictly decreasing, starting from the point $(-1, \pi)$, crossing the y-axis at $(0, \frac{\pi}{2})$, and ending at $(1, 0)$.

Inverse Tangent (Arctan): Defined as $y = \arctan(x)$ or $y = \tan^{-1}(x)$ where the domain is all real numbers $x \in \mathbb{R}$ and the range is the open interval $y \in (-\frac{\pi}{2}, \frac{\pi}{2})$. The vertical asymptotes of the tangent function become horizontal asymptotes at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$. The curve passes through $(0,0)$ and flattens out as $x$ approaches $\pm\infty$.

Composition Identities and Principal Values: The identity $f(f^{-1}(x)) = x$ always holds within the domain of the inverse. However, $f^{-1}(f(x)) = x$ only holds if $x$ is within the restricted principal range. If $x$ is outside this range, you must use the unit circle's symmetry to find the equivalent angle within the principal range (e.g., $\arcsin(\sin(\frac{3\pi}{4})) = \frac{\pi}{4}$).

Geometric Interpretation with Right Triangles: The expression $\theta = \arcsin(\frac{a}{c})$ represents an angle $\theta$ such that $\sin(\theta) = \frac{a}{c}$. This can be visualized as a right-angled triangle with an opposite side of length $a$ and a hypotenuse of length $c$. This allows for the evaluation of composite functions like $\cos(\arcsin(x))$ by finding the adjacent side length $\sqrt{c^2 - a^2}$ using the Pythagorean theorem.